Prime numbers including 1 but not 1 2 or 3

[aastrotech]

Mathematical logic. The simplest definition that describes the case is the preferred one. "Primes are any number that can be divided evenly only by itself and one". This satisfies the "simple case" requirement defining primes and includes one.

A complication added "any non sequential number that can be divided evenly only by itself and one". prohibits 1 2 and 3 from being prime. But there is no mathematical logic for the "non sequential" complication (except to illogicly prohibit 1 2 and 3 from being prime.

There is no mathematical logic to the complications of definition by adding , "whole, distinct, natural numbers" or other complications to the simple definition that defines the case of primes (except to illogicly prohibit one from being prime).

There may be a mathematicaly logical use for a complication in definition to illustrate a mathematical point, define a mathematical case or concept or illustrate a theory. I made use of a more complicated definition to illustrate this point in mathematical logic myself in paragraph two above. But this does not mean that having a use for a more complicated definition makes the more complicated definition the preferred one.

[pzkpfw]

You seem to be re-defining Primes to match your own "simplest definition".

Claiming that this "simplest definition" is therefore better, is circular.

From wikipedia:

In mathematics, a prime number (or a prime) is a natural number which has exactly two distinct natural number divisors: 1 and itself. The first twenty-five prime numbers are:

2, 3, 5, 7, 11, 13, 17, 19, ...snip...

Why do you want to re-define what Primes are?

[pzkpfw]

Note that 1 not being a Prime is important in other mathematical definitions.

If you accept 1 as a Prime, to make the definition of Prime "simpler", you would need to add "except 1" as a "complication" to those other definitions.

So you'd make one definition simpler by making others more complex.

That's not a good investment.

(The Wiki page notes some of these. e.g. see "Primality of one")

[cjameshuff]

For example, every positive integer has a unique representation as a product of integer powers of primes, and can be treated as a vector of those powers.

Primes: 2, 3, 5, 7, 11, 13...

1 = <0> = 2^0
2 = <1> = 2^1
3 = <0, 1> = 2^0 * 3^1
4 = <2> = 2^2
12 = <2, 1> = 2^2 * 3^1
15 = <0, 1, 1> = 2^0 * 3^1 * 5^1
42 = <1, 1, 0, 1> = 2^1 * 3^1 * 5^0 * 7^1

If 1 is a prime, you must specifically treat it as special and omit it from the above representation, because 1^n is always 1...there would be an infinite number of representations for each of the numbers above if 1 were considered a prime.

[Paul Beardsley]

Elegantly put, cjameshuff.

[hhEb09'1]

For example, every positive integer has a unique representation as a product of integer powers of primes, and can be treated as a vector of those powers.

Primes: 2, 3, 5, 7, 11, 13...

1 = <0> = 2^0
2 = <1> = 2^1
3 = <0, 1> = 2^0 * 3^1
4 = <2> = 2^2
12 = <2, 1> = 2^2 * 3^1
15 = <0, 1, 1> = 2^0 * 3^1 * 5^1
42 = <1, 1, 0, 1> = 2^1 * 3^1 * 5^0 * 7^1

If 1 is a prime, you must specifically treat it as special and omit it from the above representation, because 1^n is always 1...there would be an infinite number of representations for each of the numbers above if 1 were considered a prime.How do you deal with 1 = 3^0?

Your <> notation seems to imply that 1 = <0,0,0,0,...> is what is intending, and so 4=<2,0,0,...> but that's a different approach.

[cjameshuff]

How do you deal with 1 = 3^0?

Your <> notation seems to imply that 1 = <0,0,0,0,...> is what is intending, and so 4=<2,0,0,...> but that's a different approach.

No, that is indeed what I was using it to mean. I left the "0, ..." off for simplicity...I initially had them, but felt they only cluttered things up. Unstated elements in the vector are assumed to be zero, just as they are in normal place-value notation...you don't write 5 as "...05", for example.

[hhEb09'1]

No, that is indeed what I was using it to mean. Then that should read "every positive integer has a unique representation as an infinite product of integer powers of all primes" right? You have to represent 1 as 20 * 30 * 50 * 70 * .... to make the uniqueness hold.

[aastrotech]

"Primality of one")

In the wiki link states "Until the 19th century, most mathematicians considered the number 1 a prime, with the definition being just that a prime is divisible only by 1 and itself (Note it is not I redefining the definition of primes) but not requiring a specific number of distinct divisors...The change in label occurred so that the fundamental theorem of arithmetic, as stated, is valid, i.e., “each number has a unique factorization into primes". The wiki page on the fundamental theorem of arithmetic states that it is a hypothesis.(snip wiki; In common usage in the 21st century, a hypothesis refers to a provisional idea whose merit requires evaluation. )

As I said a redefinition of a definition may be useful i.e. in validaing a hypothesis but that does not make the redefinition preferred.

Preferred means preferred by a priori mathematical logic not preferred by a majority of mathematicians for convinence.(snip wiki; hypothesis old: clever idea or a convenient mathematical approach ).

The only mathematicaly logical reason for redefining a preferred definition is to show a that a nonhypothetical quandry is raised by the preferred definition.

PS Do not conclude that by making references to wiki that I am only now learning of the issues involved (a common slander posted here). I know the issues involved. That's why I know I can find them. I use wiki references to refer you to the issues involved.

[hhEb09'1]

Mathematical logic. The simplest definition that describes the case is the preferred one. "Primes are any number that can be divided evenly only by itself and one". This satisfies the "simple case" requirement defining primes and includes one.The simplest definition is the preferred one, is not mathematical logic though. It's often known as Occam's Razor, which I've argued is not even a scientific principle, because of its subjectiveness. Others disagree, but I doubt they'd call it a mathematical logic principle.
A complication added "any non sequential number that can be divided evenly only by itself and one". prohibits 1 2 and 3 from being prime. But there is no mathematical logic for the "non sequential" complication (except to illogicly prohibit 1 2 and 3 from being prime.

There is no mathematical logic to the complications of definition by adding , "whole, distinct, natural numbers" or other complications to the simple definition that defines the case of primes (except to illogicly prohibit one from being prime).You do have to include that in the definition, though, right? Else what do you do with 3.5 / .5 = 7?

[tdvance]

Prime numbers are the smallest set from which all positive integers can be generated by multiplication.

That's about as simple as the definition can be made--and including 1 makes it more complicated, requiring a "and also 1" or something similar.

Before anybody makes this mistake--1 is generated by multiplication. An empty product is by definition, 1.

[01101001]

Prime numbers including 1 but not 1 2 or 3

Please stop trying to return the definition of primes to that of the 19th century.

Think modern.

Please.

[aastrotech]

The simplest definition is the preferred one, is not mathematical logic though. It's often known as Occam's Razor,

Occam's razor referrs to solutions or explanations not definitions. From wiki; "To straightforwardly summarize the principle as it is most commonly understood, “Of several acceptable explanations for a phenomenon, the simplest is preferable.”

Often restated "the simplest solution or explanation tends to be the right one"
which, absent the equivicating term "tends", I agree is demonstrably false.

But relevant to this topic more properly stated "entities should not be multiplied unnecessarily" applies to the wiki "Primality of one" point regarding 1x1x1x3=1^3x3=3 in that it multiplies entities unnecessarily. So the preferred definition of 3 is 3=1x3=1(3) not 1x1x1x3=1^3x3=3. Although 1x1x1x3=1^3x3=3 is true it is not preferred. Multiplication (the reverse of division) is defined in terms of addition not in terms of multiplication of multiplication.


I agree that "whole" numbers could and should be logicly added to the simple definition to avoid the quandry you defined. However, absent an equivicating statement, it is usually assumed (as a mathematical convienience) that the numbers are whole when the product is required to be whole. Just as 1 times n is assumed and not written (as a mathematical convienience) for any number or expression as in 1=1x1=1(1) or 2 =1x2=1(2) or a =1xa=1(a)=1a.

PS I am obliged to point out that this discussion is getting into areas of math that could require written mathematical statements which could be written in html coding but unsupported by the BAUT forum software for some reason. For this reason I am going to have to tell you that should an answer requiring unsupported code be needed I am going to state the permitted response "I DON"T KNOW" how to answer without html coding". Also if an answer requires a criticism of the questioner's mathematical or scientific principle or premise I am going to answer "I DON'T KNOW how to answer your question without criticising your principles or premis and being fouled by a moderator here". I write this because there are statements above that do not belong in this discussion.

[pzkpfw]

Things change.

1 was considered a Prime, now it isn't. That wasn't for fun, but for valid mathematical reasons.

So the definition of Primes changed to match. 1 is no longer considered Prime.

You've admitted "whole" should be added to the "simplest" defintion you provided.

To claim that just one more word ("distinct") should not be added is simply tilting at windmills.

http://wiki.answers.com/Q/Why_is_1_not_a_prime_number
If 1 is not a prime number, then any composite number (such as 12) can be written as a product of primes in only one way (here, 2*2*3), not counting different orders. However, if 1 were a prime number, there would be infinitely many ways! We could write 12 for example, as 2*2*3, or 1*2*2*3, or 1*1*1*1*1*2*2*3. Having only one way to write a number as a product of primes is very useful when doing math.

http://mathworld.wolfram.com/PrimeNumber.html
...it requires special treatment in so many definitions and applications involving primes greater than or equal to 2 that it is usually placed into a class of its own.

http://primes.utm.edu/notes/faq/one.html

etc.

[HenrikOlsen]

PS I am obliged to point out that this discussion is getting into areas of math that could require written mathematical statements which could be written in html coding but unsupported by the BAUT forum software for some reason. For this reason I am going to have to tell you that should an answer requiring unsupported code be needed I am going to state the permitted response "I DON"T KNOW" how to answer without html coding". Also if an answer requires a criticism of the questioner's mathematical or scientific principle or premise I am going to answer "I DON'T KNOW how to answer your question without criticising your principles or premis and being fouled by a moderator here". I write this because there are statements above that do not belong in this discussion.
As this post (http://www.bautforum.com/forum-introductions-feedback/72561-poster-quoting-responding-deleted-message-post1213979.html#post1213979) mentions, it's possible to do http://bb.iaeste.org/cgi-bin/mimetex.cgi?x=\frac{-b\pm\sqrt{b^2-4ac}}{2a} and much more, though it requires you to know TeX rather than html.
It's using my server and members of this forum has explicit permission to use it for formula display here.

If you only need sub/superscript, use the [ sub ][ /sub ] and [ sup ][ /sup ] tags without the spaces.

[aastrotech]

it's possible to do http://bb.iaeste.org/cgi-bin/mimetex.cgi?x=\frac{-b\pm\sqrt{b^2-4ac}}{2a} and much more, though it requires you to know TeX rather than html.
It's using my server and members of this forum has explicit permission to use it for formula display here.

If you only need sub/superscript, use the [ sub ][ /sub ] and [ sup ][ /sup ] tags without the spaces.

Interesting. Thank you. I will consider it for another discussion. But doesn't this require hotlinking from another site prohibited by the rules?

This discussion has I believe run its course as another moderator has made a prosaic argument rather than a mathematical one based on an out of context misquote of my point to a standard rule of mathematics and I DON'T KNOW how to answer his argument without criticising his principles and being fouled by a moderator here. I am, however, happy not to criticise someone's principles.

And thank yous where deserved for an admirable argument.

[pzkpfw]

If that was me, feel free to PM me your response.

Openly (i.e. in public) stating: No harm no foul from it's content.


P.S. I should add - if you think an offence was committed even by a moderator you can use the report triangle.

[hhEb09'1]

misquote A moderator made a misquote? I'm sure it can happen, but I don't find it in this thread. Can you be specific?

[aastrotech]

If that was me, feel free to PM me your response.

Openly (i.e. in public) stating: No harm no foul from it's content.


P.S. I should add - if you think an offence was committed even by a moderator you can use the report triangle.

I DON'T KNOW how to answer your any of these points without criticising your principles and being fouled by a moderator here.

I hope that claifying what "I don't know" to reflect the truth is allowed here because ""I don't know" an answer to your points" is not true.

[aastrotech]

A moderator made a misquote? I'm sure it can happen, but I don't find it in this thread. Can you be specific?

Not without critcising his principles and possibly (likely) being fouled by a moderator. Sincerely, I am sorry. Although specifying and criticising may be informative for the reader which is one reason for being here I DON'T KNOW how without criticising principles and being fouled by a moderator here.

[hhEb09'1]

Not without critcising his principles and possibly (likely) being fouled by a moderator. Sincerely, I am sorry.Then are you saying it is a deliberate misquote? I only see a couple posts that even directly quote your posts, and I don't see a misquote. Do you mean a paraphrase?

[pzkpfw]

Not without critcising his principles and possibly (likely) being fouled by a moderator. Sincerely, I am sorry. Although specifying and criticising may be informative for the reader which is one reason for being here I DON'T KNOW how without criticising principles and being fouled by a moderator here.

I'm not sure why you are so nervous. If you have a genuine point it will be dealt with.

You can do this privately:
* via reporting the post (that brings it to the attention of all moderators).
or by
* sending (me?) a PM.

I tried to ease your mind by pointing out (in public) "no harm no foul". I'm not trying to bait you into committing a "crime". I'm a "grown up", and unless your PM was simply a rant filled with invective (which I don't expect you to send) I'm not going to get "hurt" and try to "punish" you in revenge.

I am inviting the PM, so we can sort this issue out.

P.S. 1: More than one member replied to this thread. You could deal with those other replies rather than abandoning the thread due to one particular member.

P.S. 2: Moderators are "normal" members too. Any posts posted by a moderator that are not obvious "moderator posts" (one reason we use colour) should be treated as from any other member. i.e. if you think a moderator is wrong, you can say so. I for one am happy to have my knowledge or logic corrected.

[aastrotech]

I guess some of you don't know what "I DON'T KNOW how to answer your question without criticising your principles and being fouled by a moderator here" means. It means that discussion of questions that I don't know how to answer without criticising principles and being fouled by a moderator here is over as far as I'm concerned.

Maybe this will help.

Ask your question or make your statement and we'll see if I do know how to answer your question without criticising your principles and being fouled by a moderator here".

The well of reasonable points on this topic while getting low has not run dry. It might be illustrative for some to look up the list of types of primes (there are over 500) some of which require the preferred definition of prime where 1 is prime most of which require the revised definition of prime where 1 is not prime. But this is not the only point that might illuminate this topic.

[Jens]

This is getting quite mysterious.

But on topic, and on a lighter note: why not just give a new name to the set of numbesr including the primes and 1? I would suggest prime'.

[pzkpfw]

I guess some of you don't know what "I DON'T KNOW how to answer your question without criticising your principles and being fouled by a moderator here" means.

Oh goodness.

I tried to make it clear that:
1. I can handle your questioning of my principles or whatever. [It won't make me cry and it won't make me suspend or ban you!]
2. You can do it in private if you wish.
3. I have invited you to make your claims to me directly by PM, if you wish. [This is not a "trap".]
4. I have also suggested reporting a post that you wish to complain about. [So all moderators can look at it.]
And in summary:
5. Due to 1, 2, 3 & 4 above you will NOT be "fouled by a moderator". [Even if no-one agrees with you, if it's an honest opinion why would anyone "foul" you?]

I don't know what more can be done to resolve this issue, whatever it is.

Cheers,

[pzkpfw]

Back on topic:

It might be illustrative for some to look up the list of types of primes (there are over 500) some of which require the preferred definition of prime where 1 is prime most of which require the revised definition of prime where 1 is not prime.

This is an interesting point, but it might be more useful for you to back up your contention made in post 1 by providing such examples, rather than asking us all to search for them.

Did you mean to write: "...most of which require the revised definition of prime where 1 is not prime"? (My underline)

If so, then this would indicate that there would be fewer cases where the definition of a type of Prime must say "including 1" (in the current 1-not-a-Prime standard) than there would be cases where the definition of a type of Prime had to say "not including 1" (in the older 1-is-a-Prime standard).

This would still seem to back-up the preference for 1 not being a Prime.


Added later:
But on topic, and on a lighter note: why not just give a new name to the set of numbesr including the primes and 1? I would suggest prime'.

How about "Primes++" ? (Programming joke)

[slang]

I would suggest prime'.

Subprime.

[HenrikOlsen]

Interesting. Thank you. I will consider it for another discussion. But doesn't this require hotlinking from another site prohibited by the rules?
As with all our rules it's there for a reason and that reason rather than the text of the rule is what matters.

The "no hotlinking" rule is about not using bandwidth from other sites without their permission.
This is why sites that are specifically for the purpose of hosting files for inclusion in other sites are excepted from the rule, as are any site where explicit permission is given.

[Paul Beardsley]

Considering some of the content of the previous thread about one being/not being a prime, I was curious to see the direction this thread would take.

So far, it doesn't seem to be going anywhere.

[Ivan Viehoff]

The concept of primes is found in ring theory, the integers (under multiplication and addition) being a special case of a ring. Our concept of prime numbers today is aligned to the general concept of primeness in ring theory.

In rings, there are elements known as "units". Units divide everything. In the case of the integers, there are two units, 1 and -1. Observe that 1 = (-1)^2 and -1 = (-1)^3, so divisibility by a unit is trivial and needs to be excluded from factorisations in general. Observe also that -2 is just as prime as +2, but they are in effect the same prime number, since one is just the other multiplied by a unit, so we need to take the units out of all of this. So, units are not primes. In the ring of complex integers, numbers of the form a+ib where a and b are integers, 1, -1, i and -i are all units.

The definition people are giving - a number only divisible by a unit and itself - is in ring theory the definition of an irreducible element, not a prime element. Primes are in general a subset of irreducibles. It so happens that in the integers the irreducibles and primes are the same, because, among other things, the integers are a unique factorisation domain, and in ufds primes and irreducibles are the same. The complex integers are not a unique factorisation domain - you can sometimes factor an element by more than one distinct way by irreducible elements. So in the complex integers, the irreducibles and the primes are not the same.

In general ring theory, a prime element is the generator of a prime ideal, which you can read about here:
http://mathworld.wolfram.com/PrimeIdeal.html
In the case of the integers, the ideal 2Z is the same as the ideal (-2)Z, so that gets the units out of the way.

A simpler way of putting the above definition is that a prime element is an element (not a unit) such that p divides ab implies that p divides at least one of a and b. But this is a definition too complicated for introducing people to prime integers.

[jfribrg]

Fundamental Theorem of Arithmetic is the basis of all of this prime and composite stuff. This states that every natural number > 1 can be expressed as a product of prime numbers in one and only one way except for order.
For example: 5 is prime

If 1 is considered a prime, then 5 = 5 x 1 and 5 = 5 x 1 x 1 which is two different ways of expressing 5 as a product of primes, and as a result, the Fundamental Theorem of Arithmetic is wrong and there is not such thing as prime or composite. If you argue that 1 is somehow special, and make special rules to make sure that 1 does not interfere with all of the theorems that depend on the Fundamental Theorem, then what you have really done is exactly what the mathemeticans did centuries ago: Define 1 as neither prime nor composite. It is called Unary.

Now, what I want to know is whether 0.99999999..... is prime or not :)

[aastrotech]

But on topic, and on a lighter note: why not just give a new name to the set of numbesr including the primes and 1? I would suggest prime'.

I'm going to use their original titles; "The Preferred Definition of Primes" and "The Revised Definition of Primes".

There is nothing wrong with revising definitions. We do it all the time. It's allowed in the rules of mathematics as long as the revision follows the rules of mathematics other than the exeption defined by the revision and the revision itself doesn't produce a mathematical quandry. The exact definition of Pi is incalculable and almost useless. Approximate definitions of Pi are used all the time. Just don't make the mistake of thinking a revised definition is the preferred definition. This is not a hard and fast rule. Sometimes a revised definition by mathematical logic becomes the preferred definition. But not just because a revised definition is usefull.

[pzkpfw]

I'm going to use their original titles; "The Preferred Definition of Primes" and "The Revised Definition of Primes".

There is nothing wrong with revising definitions. We do it all the time. It's allowed in the rules of mathematics as long as the revision follows the rules of mathematics other than the exeption defined by the revision and the revision itself doesn't produce a mathematical quandry. The exact definition of Pi is incalculable and almost useless. Approximate definitions of Pi are used all the time. Just don't make the mistake of thinking a revised definition is the preferred definition. This is not a hard and fast rule. Sometimes a revised definition by mathematical logic becomes the preferred definition. But not just because a revised definition is usefull.

Sorry, but I don't see where it was shown that the "preferred" defintion of Primes is the "original" one (where 1 is included).

[hhEb09'1]

Sorry, but I don't see where it was shown that the "preferred" defintion of Primes is the "original" one (where 1 is included).It's not. I just opened a copy of Euclid's Elements, from the 3rd Century BC (translation, Heath, Dover). He doesn't even consider 1 a number, he calls it a unit. Apparently, that was true of others as well, before him, like Aristotle. Some of them didn't consider 2 to be a prime number either, not sure why.

ETA: Euclid's Prop. 14, Book IX, is considered essentially the fundamental theorem of arithmetic, that "a number can be resolved into prime factors in only one way" (from the footnote).

[Paul Beardsley]

Sorry, but I don't see where it was shown that the "preferred" defintion of Primes is the "original" one (where 1 is included).

As I demonstrated in the other thread, the definition of prime number that explicitly excludes one appears in the Chambers Dictionary, the Penguin Dictionary of Mathematics, and the material provided for Open University mathematics courses.

I have not seen any formal definitions that include one. But then, I don't have any dictionaries that were published in the 19th century.

[pzkpfw]

It looks like we all agree then.

That is, using the revised definition of "all agree", where 1 is excluded.

[aastrotech]

I think I've thought of a way to simplify this problem.

A set of numbers is defined. The definition conforms to the rules of mathematical logic. A subset is defined with a revision of the definition of the first set. The definition in the revision prohibits one of the numbers in the first set from being part of the second set. That is why the second set is a subset. All the rest of the numbers in the subset are the same as the larger set. The subset of that set is discovered to have a great deal of utility. It has so much utility that for convienence it is given the name of the larger set. The first set still exists. Nameless...homeless...for all time...Sniffle.

So this whole issue is not an issue of mathematics. It is an issue of the name. It is a semantic issue.

[pzkpfw]

Close, but not quite.

In the case of Primes, the "subset" is considered to be the set. That is due to entirely mathematical reasons, as explained on page 1 (and page 2) of this thread, by multiple posters.

It's not just a case of giving the set of "Primes without 1" the name of "Primes"; it is the currently accepted meaning of Prime; again, for valid mathematical reasons.

It's not just semantics. The number 1 is no longer considered to be a Prime number - whatever we call the set of Primes.

[Jens]

Actually, I think that it is really a question of semantics. Because the issue is how we define something. But I don't see any reason to sniff about it. In Greek the word "aster" was (I think) used to refer to all heavenly bodies. Now the word "star" in English is restricted to what we now understand to be a unified set, luminous stars. So the term is no longer used that way, and if want to refer to everything outside of the atmosphere we can say "celestrial objects" or something like that. So similarly, the word for the set with the primes and 1 was moved to the set without the 1 for a similar reason, only there is no real need to refer to the set plus 1. But it's nothing to mourn, it's just a word changing meaning.

Take a word like "salary," which was originally used to refer to the salt that soldiers were paid. Now we use it to mean any payment in general. I don't think anybody mourns the loss of the other meaning.

[pzkpfw]

So similarly, the word for the set with the primes and 1 was moved to the set without the 1 for a similar reason, only there is no real need to refer to the set plus 1.

I don't think anyone dissagrees that the set of Primes used to include 1 but now doesn't; so that what specific meaning is conveyed by "Prime" has changed.

But post #1 of this thread tried to argue that the "simpler" definition* of "Prime" (that includes 1) was "preferred". By extension, this implies that Specifically, post #1 argued that 1 should still be considered a Prime (my underline):

The simplest definition that describes the case is the preferred one. "Primes are any number that can be divided evenly only by itself and one". This satisfies the "simple case" requirement defining primes and includes one.

So it (this thread) is not just a matter of semantics, it's about what a Prime is.

Note aastrotech still (post #37) argues (my underline)...:
... It has so much utility that for convienence it is given the name of the larger set. The first set still exists. ...


(* In contrast, the consensus is that this "simpler" definition is wrong anyway - as it includes 1; which isn't a Prime.)

[Ivan Viehoff]

Now, what I want to know is whether 0.99999999..... is prime or not :)
I realise you were being funny, but in case anyone else is wondering, it isn't a prime. There are no primes in the real numbers, because anything factors anything else in the real numbers. The concept of primality of numbers exists in integer arithmetic. There are no fractions or decimals in the integers, so we don't have to worry about them. The rationals and reals respectively form a field - in a field every non-zero element has a multiplicative inverse, so every non-zero element is a unit, and none of them are primes.

For amusement, observe that 2 is neither prime nor irreducible in the complex integers:
2 = (1+i)(1-i) = (-i)(1+i)(1+i)

[hhEb09'1]

I don't think anyone dissagrees that the set of Primes used to include 1 but now doesn't; so that what specific meaning is conveyed by "Prime" has changed.Actually, I offered evidence that the set of primes originally didn't include 1, going all the way back to Aristotle. :)

[SeanF]

It looks like we all agree then.

That is, using the revised definition of "all agree", where 1 is excluded.
I gotta tell you, the humor in this post is downright beautiful. :clap:

[geonuc]

I gotta tell you, the humor in this post is downright beautiful. :clap:
I think it would make a good skit for Family Guy. :lol:

[mike alexander]

I think it would make a good skit for Family Guy. :lol:

Where it ends with Chris calling everyone a poopy-head and running upstairs?

[hhEb09'1]

oooo can I be Chris??

[aastrotech]

But post #1 of this thread tried to argue that the "simpler" definition* of "Prime" (that includes 1) was "preferred". By extension, this implies that Specifically, post #1 argued that 1 should still be considered a Prime (my underline):
Incomplete. I said that by "the simple definition" under the terms of mathematical logic 1 is prime whatever "the consensus" currently, (fashionably) calls it.
I don't see how it's not an issue of semantics.

Does the simple definition for the set of numbers formerly called prince oops I mean prime, create a mathematical quandry that renders it mathematicly wrong to define a set of numbers that way?

Hey, wait a minute, have I discovered a name for the set of numbers formerly called prince? We could call them "princely numbers". Hey, I call dibs. I named it.

So would you agree that by the definition of numbers called prince that 1 is a princely number?

Hey, wait a minute, maybe that symbol Prince carried and wanted to be known as somehow means prime.

[Paul Beardsley]

Does the simple definition for the set of numbers formerly called [snip "joke"] prime, create a mathematical quandry that renders it mathematicly wrong to define a set of numbers that way?
This question has been answered repeatedly.

[hhEb09'1]

In the wiki link states "Until the 19th century, most mathematicians considered the number 1 a prime, with the definition being just that a prime is divisible only by 1 and itself (Note it is not I redefining the definition of primes) but not requiring a specific number of distinct divisors...The change in label occurred so that the fundamental theorem of arithmetic, as stated, is valid, i.e., “each number has a unique factorization into primes". The wiki page on primes, down to the paragraph quoted. (http://en.wikipedia.org/wiki/Prime_number#Primality_of_one)

Hmmm, maybe this wiki needs some edits.... I wonder why it doesn't mention Aristotle? :)

[Perikles]

It's not. I just opened a copy of Euclid's Elements, from the 3rd Century BC (translation, Heath, Dover). He doesn't even consider 1 a number, he calls it a unit. Apparently, that was true of others as well, before him, like Aristotle. Some of them didn't consider 2 to be a prime number either, not sure why.Presumably because it was widely held that 3 was in fact the smallest actual number, 'the first one to have hight, width and length' (Theon Phil. util. math. 46.14.) The author of my quotation is as as late as 2nd Century AD. I have been trying, and failing, to understand the Greek concept of numbers for some time.

[hhEb09'1]

Theon And apparently Nicomachus. They both seemed to have held 3 as the first prime number. Aristotle, 2 though.

[Perikles]

And apparently Nicomachus. They both seemed to have held 3 as the first prime number. Aristotle, 2 though.Yes, I've just found the reference in Aristotle (Topica 157b1) and a page from Heath, History of Greek mathematics here (http://books.google.es/books?id=drnY3Vjix3kC&pg=PA73&lpg=PA73&dq=Aristotle+prime+number+two&source=bl&ots=v0Z_vwHoh5&sig=AL_5N-rPn_c-K1JAKZrgOUT5lro&hl=es&ei=BD9KSrvoBJiQjAfLl5Rj&sa=X&oi=book_result&ct=result&resnum=4) which summarizes nicely.

[pzkpfw]

Incomplete. I said that by "the simple definition" under the terms of mathematical logic 1 is prime whatever "the consensus" currently, (fashionably) calls it.

But the "simple definition", if it includes 1, is wrong.

I understand why you prefer that defintion, and your point about 'simpler being better'... but a defintion is of no use if it gives the wrong result.

For valid mathematical reasons (explained in this thread) 1 is not considered to be a Prime number.

That is not "fashion", any more than the currently best-known value for pi. That is the current "new improved" mathematical understanding. So your simpler definition is incorrect.

In the end, your only argument so far for 1 being a Prime seems to be so that the definition of Prime can be simpler. That's circular.

not be a Prime is useful in more "types of primes" than if 1 were Prime. So even by your "simpler is best" concept - 1 should not be considered Prime.]


I don't see how it's not an issue of semantics.

If it were only semantics, we'd not be arguing whether 1 is a Prime or not. I don't care if you call them Primes, Prince or Apple pie. They don't include 1, and your "simpler defintion" does not work.

[aastrotech]

pzkpfw,

Are you saying that the definition of a set of numbers "all whole numbers divisible only by the whole numbers of themselves and one to yield a product of whole numbers" doesn't contain in the set the number one? That the number 1 cannot be divided by itself? That defining a set of numbers this way will be called circular reasoning and fouled by a moderator?

[pzkpfw]

No, I'm saying that 1 is not a Prime.

You can come up with what you think is a definition of Primes, and that definition may allow 1; but that is not a definition of Primes, because 1 is not a Prime.

For example, you can say that the definition of Prime numbers is:

"all whole numbers divisible only by the whole numbers of themselves and one to yield a product of whole numbers"

...and yes, that set would contain 1.

But 1 is not a Prime, so that definition is flawed. (It may have been considered correct in, say, 1950, but not now.)

You can certainly define a set of numbers as you do above, and that set will contain 1 because 1 is divisible by 1. Just don't call that the set of Primes. (Or try to argue that because of your defintion that 1 is a Prime).

(A pedant might argue that the word "and" in your definition implies exclusivity, so that the number and 1 must be distinct values, but I'll give you this one.)

----

....and that any attempt to do so will be called circular reasoning and fouled by a moderator?

I really don't get what it is that you are so nervous about here. If you think I'm wrong, say so, and try to prove it. No moderator will foul you for that. I've been telling you that for days now.

[robross]

Now, what I want to know is whether 0.99999999..... is prime or not :)

Well, 0.99999999... = 1, so by definition, 0.99999999... is not prime. QED.

Rob

[aastrotech]

Well I think this discussion has reached a conclusion. I'm satisfied with the results. I think the issue has been clearly delineated. I don't know of anything more to add.

Thanks again, where warranted, for participating...reasonably.

[Paul Beardsley]

I don't know of anything more to add.
Except the apology you owe me.

[jfribrg]

Well, 0.99999999... = 1

Maybe we should create another thread to discuss this in greater detail :)

(Runs and hides before anyone can throw a tomato).

[robross]

Maybe we should create another thread to discuss this in greater detail :)

(Runs and hides before anyone can throw a tomato).

What is there to discuss? Are you claiming there is some debate on this point?

Let x = .999...

Then 10x = 9.999...

Remembering it is legal to subtract the same value from both sides of an equation:

10x = 9.999...
- x = .999...
-------------------
9x = 9

x = 1, QED.

What did you want to discuss about this?

Rob

[aastrotech]

prime
First in excellence, quality, or value. See Usage Note at perfect.
First in degree or rank; chief. See cardinality
First or early in time, order, or sequence; original.

perfect
Lacking nothing essential to the whole; complete of its nature or kind.
Being without defect or blemish: a perfect specimen.
Completely corresponding to a description
Accurately reproducing an original
Complete Pure; undiluted; unmixed

primitive
Not derived from something else; primary or basic
Of or relating to an earliest or original stage or state
Characterized by simplicity

[pzkpfw]

None of which proves 1 is a "Prime number" in contrast to current mainstream mathematical theory; though it does perhaps show where the name "Prime number" derives from.

The relationship of those defnitions to the meaning "Prime number" can't be as simplistic as you imply*, else they'd not explain others like 5 and 7.

1 is thought of as a "Unit". Something slighter deeper than "Prime".


(* I assume you use those quotes to try to back up your "1 is a Prime number" stance. Correct me if I'm wrong.)

[aastrotech]

None of which proves 1 is a "Prime number"

Correct as to the revised definition of "Prime Number" (caps). But is in dispute of the contension that "Prime Numbers" (caps) are "prime" (no caps) numbers.

[pzkpfw]

Um, now I'm lost as to the purpose of this thread. Your OP was this (my underline and bold):

Mathematical logic. The simplest definition that describes the case is the preferred one. "Primes are any number that can be divided evenly only by itself and one". This satisfies the "simple case" requirement defining primes and includes one.

A complication added "any non sequential number that can be divided evenly only by itself and one". prohibits 1 2 and 3 from being prime. But there is no mathematical logic for the "non sequential" complication (except to illogicly prohibit 1 2 and 3 from being prime.

There is no mathematical logic to the complications of definition by adding , "whole, distinct, natural numbers" or other complications to the simple definition that defines the case of primes (except to illogicly prohibit one from being prime).

There may be a mathematicaly logical use for a complication in definition to illustrate a mathematical point, define a mathematical case or concept or illustrate a theory. I made use of a more complicated definition to illustrate this point in mathematical logic myself in paragraph two above. But this does not mean that having a use for a more complicated definition makes the more complicated definition the preferred one.

Your OP was clearly about whether the set of prime numbers (which we all clearly knew meant 1, 2, 3, 5, 7, 11, ...) included 1 or not.

Edit to add: Your arguement being based on "simpler definition" = "preferred definition".

The discussion in this thread has largely been about this topic.

Now you say:

But is in dispute of the contension that "Prime Numbers" (caps) are "prime" (no caps) numbers.

Are you seriously trying to re-define your argument into some quibble on what we mean when we say "Prime Numbers" (caps) versus "prime numbers" (no caps)?

If you really wanted to discuss what the "prime" or "Prime" means in "Prime Number" or "prime number" or whatever, why did you wait until post #63 to say so?!

What exactly is your point?

Post #61 contained some defintions, but no commentary from you on what they were intended to prove or demonstrate or show. Try being clearer.

[aastrotech]

It's a mudpie. I don't care how much it looks like a pie, how shiney and creamy it looks, how much gold went into it, how many mathematicians call it a pie or how many picadores with their barbs or matadores with their swords insist that I eat it. I know it's a mudpie and I'm not taking a bite.

[baric]

The simplest definition that describes the case is the preferred one. "Primes are any number that can be divided evenly only by itself and one"

Keeping in theme with the original poster's desire that "simpler is better", I suggest that we change the definition to "Primes are any number".

That is simpler and therefore must be a better definition. :)

[Paul Beardsley]

What exactly is your point?
It's a clear case of aastrotech trying to redefine a key part of mathematics rather than admit he owes me an apology for his abusive comments in his "Marking the moon" thread.

[Jens]

Are you saying that the definition of a set of numbers "all whole numbers divisible only by the whole numbers of themselves and one to yield a product of whole numbers" doesn't contain in the set the number one?

Sorry to kind of flog a dead horse, but I'm not so sure anymore that this contains 1. Wouldn't it depend on whether the "and" is seen as inclusive or exclusive. For example, if you say "I bought an apple AND a banana," it has to be two things, but when you say "he is a painter and an architect" it's talking about the same person. If you took the "and" to be a very strong "and" so that it would read something like "divisible only by the whole numbers of themselves as well as one," then I think 1 could be excluded.

[baric]

Sorry to kind of flog a dead horse, but I'm not so sure anymore that this contains 1. Wouldn't it depend on whether the "and" is seen as inclusive or exclusive. For example, if you say "I bought an apple AND a banana," it has to be too things, but when you say "he is a painter and an architect" it's talking about the same person. If you took the "and" to be a very strong "and" so that it would read something like "divisible only by the whole numbers of themselves as well as one," then I think 1 could be excluded.

The precise wording of the definition of prime numbers is not so important as the recognition of what they are, which is where the OP strays. For number theorists, the listing of primes is akin to the periodic table of elements. Primes are the "building blocks" of natural numbers in that they represent the minimum set of natural numbers than can be used, through multiplication, to represent the whole set.

The number of words or the grammatical structure of the definition needed to convey this concept is unimportant. Who cares? Those are just imprecise words from a disorganized language!

[aastrotech]

what they are, which is where the OP strays. For number theorists, the listing of primes is akin to the periodic table of elements. Primes are the "building blocks" of natural numbers in that they represent the minimum set of natural numbers than can be used, through multiplication, to represent the whole set.


"The whole set"?..except for one...right? Is one not a natural number now? Not to be included in the set of natural numbers? What set of revised Primes can be used through multiplication to represent the number one? Seems to me these building blocks are not to be living up to their design specs.

[HenrikOlsen]

It's a clear case of aastrotech trying to redefine a key part of mathematics rather than admit he owes me an apology for his abusive comments in his "Marking the moon" thread.
This is the second time you've tried to derail this thread with your personal problems with aastrotec.
They don't belong here, don't bring them up again.

[HenrikOlsen]

"The whole set"?..except for one...right? Is one not a natural number now? Not to be included in the set of natural numbers? What set of revised Primes can be used through multiplication to represent the number one? Seems to me these building blocks are not to be living up to their design specs.
1=2^0, case closed

[aastrotech]

1=2^0, case closed

I'm afraid I'm not famliar with that. Could you expand?

[Nick Theodorakis]

I'm afraid I'm not famliar with that. Could you expand?

Anything except zero raised to the zeroth power equals one. See this discussion (http://oakroadsystems.com/math/expolaws.htm), or this wiki link (http://en.wikipedia.org/wiki/Exponentiation#Exponents_one_and_zero), for example.

Nick

edit: apparently the value of 00 is contentious (http://en.wikipedia.org/wiki/Exponentiation#Zero_to_the_zero_power).

[baric]

"The whole set"?..except for one...right? Is one not a natural number now? Not to be included in the set of natural numbers? What set of revised Primes can be used through multiplication to represent the number one? Seems to me these building blocks are not to be living up to their design specs.

This is exactly why I used the phrase "imprecise words in a disorganized language". Numbers are what they are regardless of how we try to describe them. One is the base unit for integers. Primes are the multiplicative building blocks for natural numbers. There is no amount of semantic quibbling that is going to change those basic facts.

[tdvance]

"The whole set"?..except for one...right? Is one not a natural number now? Not to be included in the set of natural numbers? What set of revised Primes can be used through multiplication to represent the number one? Seems to me these building blocks are not to be living up to their design specs.

As I said before, 1 is the empty product by definition, just as 0 is the empty sum, True is the empty conjunction, False is the empty disjunction, a singleton is an empty Cartesian product, 0 is an empty linear combination, and so on.

[robross]

Is there an actual point being made in this thread? I'm not familiar with the rules of "Off-Topic Babbling", so maybe there doesn't need to be a point.

But I'm not sure if the OP is stating that 1 *should* be considered a prime, or if he is just stating it *used* to be a prime and now it's not. If it's the later, then I agree with him. If it's the former, I think that would break a lot of existing mathematics and I don't see any added benefit.

Math evolves like anything else, and 1 has evolved in our understand to be not-prime. Another historical change for contemplation: zero at one time was not a number either, but it's become pretty darned useful and I wouldn't want to give it up. Would you?

Rob

[pzkpfw]

It may be helpful to see where this thread was spun off from: http://www.bautforum.com/space-exploration/89417-marking-moon-message-future.html

Edit to add: that's just for reference and background to this thread. That other thread is locked, please don't anyone revisit it here. Thanks.

[aastrotech]

1=2^0, case closed

Looking into this finds at its core the convention not fact that "anything multiplied by 1 is no multiplication at all".

Understandable as a convention (or convienience) in some cases. "Anything multiplied by 1 is no multiplication at all" strikes at the core of multipication. Since multiplication is defined in terms of addition it tells how many multiplicands the multiplier gives (or adds to) the product. E.g. the multiplicand x is given by the multiplier y to the product xy y number of times for any number x y. For x=1 and y=1 and xy=xy. 1 is given by the multiplier 1 time to the product to equal 1. for x=3 and y=5 and xy=15 3 is given by the multiplier 5 times to the product to equal 15.

(from wiki) In "Euclids lemma" (a proof of the fundamental theorem of aritmatic) a lemma is "anything which is received, such as a gift, profit, or a bribe" and there is no formal distinction between lemma and theorem, only one of usage and convention.

In the wiki fundamental theorem of aritmatic "There are natural extensions of the hypothesis of this theorem, which allow any non-zero integer to be expressed as the product of "prime numbers" and "invertibles". For example, 1 and -1 are allowed to be factors of such representations (although they are not considered to be prime)."

Which is not only a contradiction in terms but also a bit of circular reasoning.

[agingjb]

Hmm. OK, let's change the convention back. "1 is a prime."

But we need a term for the set of prime natural numbers greater than 1, a set of numbers that plays a important part in the theory of numbers (such an important part that the convention of excluding 1 from the set of primes was found convenient). What do we call this set?

[aastrotech]

Hmm. OK, let's change the convention back. "1 is a prime."

But we need a term for the set of prime natural numbers greater than 1, a set of numbers that plays a important part in the theory of numbers (such an important part that the convention of excluding 1 from the set of primes was found convenient). What do we call this set?

How about "Suspect Primes" to remind the user that any theory or assumption derived from them is suspect?

[Jeff Root]

This is my outsider's take on the controversy:

A prime number is a positive integer which is not the product of
other integers. However, it turned out that it is often convenient
to omit the number 'one' from the definition of 'prime number', so
the exception of the number 'one' was tacked on to the definition,
even though it doesn't naturally fit there.

-- Jeff, in Minneapolis

[Ivan Viehoff]

To do mathematics, we have to make precise definitions, otherwise our proofs will be sloppy and won't work. For example around about the early 19th century, someone pointed out that according to the definition of regular polyhedron current at the time, there were in fact 7 of them, not just the 5 platonic solids, but the additional two have intersecting faces. The long-standing proof that there could be only 5 was in fact invalid according to the stated definition. Also Euler's Theorem (for counting faces edges and vertices) only works if you are careful how you define a more general polyhedron; the early definitions were too general and the early proofs of Euler's theorem faulty. So precise definitions are required to do mathematics.

In some cases we can make choices in our definitions. For example: A. The natural numbers can start at 0 or 1 according to taste. It becomes a matter of convention, and in that case there is not a settled convention. People who start natural numbers at 0 often have to say "non-zero natural number"; those who start them at 1 often have to say "or zero". B. It is conventional to include equality in the subset relationship. So if we are only interested in proper subsets, we have to say so. The alternative convention of making subset only apply to proper subsets could have applied, and if we were interested in subset or equal, we could then have made that clear. In this case, there is a settled convention.

Likewise, we could in principle have settled either way with primes: if we included units we could then explicitly exclude them on a case-by-case basis. It is just like "subset".

There are theorems about "odd primes", ie primes excluding 2. We could redefine "prime number" to exclude 2, as was once current apparently, so that we would no longer have to talk about "odd primes" in that context. But more widely that would be perverse, as a definition of "prime" that excluded 2 would have no counterpart in the more general theory of prime elements of rings - we would now need a different word for primes in general ring theory.

Most of our present theorems about primes do not extend to units, so on balance it is much more convenient to define primes to exclude units. That is why it is a settled convention that primes exclude units. But at the end of the day it is only a convention. If "primes plus units" were in general a useful category, mathematicians would probably have invented a convenient word or phrase for it.

[Jeff Root]

Which were invented first? Prime numbers or negative integers?

-- Jeff, in Minneapolis

[Perikles]

Which were invented first? Prime numbers or negative integers?

-- Jeff, in MinneapolisPrime numbers (probably), but we could argue as to whether they were invented or discovered.

[agingjb]

How about "Suspect Primes" to remind the user that any theory or assumption derived from them is suspect?

I think the choice of the word "suspect" does much to persuade me that the present convention, that 1 is not a prime, is preferable.

[aastrotech]

It may be helpful to see where this thread was spun off from: http://www.bautforum.com/space-exploration/89417-marking-moon-message-future.html


The only thing that I see from that thread as useful to this thread is to point out that the post that triggered this exposition; "one is not a prime", might have been more accurate had it said "one is not a conventional prime".

On that token the title "conventional prime" rather than "suspect prime" still carries a caveat albeit less overt.

[aastrotech]

I like this;

Euclid of Alexandria, when asked by one of his students of what use was the study of geometry, asked his slave to give the student threepence, "since he must needs make gain of what he learns."

I expect some students got more value from what they learned than threepence.

I especially like the idea that he told his slave to do it. He didn't have to specify which student it was directed at. I wonder if the student who asked the question stood up to claim his reward or if the slave got to keep the money.

[01101001]

On that token the title "conventional prime" rather than "suspect prime" still carries a caveat albeit less overt.

You keep trying to tread water by grabbing at words.

Your old idea is dragging you down.

Glub.

[robross]

You keep trying to tread water by grabbing at words.

Your old idea is dragging you down.

Glub.

I think we all go through phases like this, but most of us are humble enough to admit we were wrong, learn from our mistakes, and move on.

I once got the notion that I would free mathematics of an inconsistency it had to endure for centuries - I wanted to define 1/0. Alas, it was a fruitless attempt as I quickly realized allowing this led to more problems than any possible benefit. As others had already tried long before I did.

So I conceded that 1/0 truly must remain undefined. It would be nice if the OP would do a little research and discover *why* 1 must remain not-prime.

Rob

[tdvance]

You CAN define 1/0, but you have to give up other nice things to do so. Consider, for example, the Riemann Sphere, which is made out of the complex plane an an additional point called "infinity" (so with the right definition of distance (not the usual Euclidean distance of complex numbers), so infinity is a "finite" distance from 0 (!!!) it is metrically equivalent to a sphere) so that functions of the form (az+b)/(cz+d) (Mobius functions), for ad-bc != 0, are defined everywhere on the sphere. You are unable to compute some things, though, like infinity - infinity.

[HenrikOlsen]

How about "Suspect Primes" to remind the user that any theory or assumption derived from them is suspect?
No theory or assumption based on them can be suspect unless there's disagreement on what set is meant by the name and all current mathematicians
understands primes of the natural numbers to exclude one.

This is basically because it's easier to refer to "the primes and one" in the few cases where that's relevant than it is to refer to "the primes excluding one" most of the time.

Once you get it into your head that names are a matter of convenience more than a matter of fitting some arbitrary definition of simplicity, this whole mess stops being an issue.

Unless of course it's all about you being unable to back up one single step and note that 1, 4, 9 are the squares of the first 3 natural numbers, which would have made that other thread about something interesting rather than the useless quibble this one is too.

[Merkin Muffley]

You are unable to compute some things, though, like infinity - infinity.

And that's the way it has to be - you make infinity a number, you lose some properties of numbers.

[pzkpfw]

Unless of course it's all about you being unable to back up one single step and note that 1, 4, 9 are the squares of the first 3 natural numbers, which would have made that other thread about something interesting rather than the useless quibble this one is too.

I was wondering when someone would mention this. As it was a claim in the "marking the Moon thread" it had made me think of the monoliths in "2001 a space odyssey". I assumed that was the inspiration of the idea posted. According to the Wikipedia, the 1 4 9 ratio of the monoliths is the squares of the first three "integers".

This morning I dug out my copy of the book, as I wanted to see if back when it was written Arthur C. Clarke had used "integers" or "Primes" - and in it (my printing by Arrow, dated 1978 (I think)) he does use "integer".

[pzkpfw]

In the wiki fundamental theorem of aritmatic "There are natural extensions of the hypothesis of this theorem, which allow any non-zero integer to be expressed as the product of "prime numbers" and "invertibles". For example, 1 and -1 are allowed to be factors of such representations (although they are not considered to be prime)."

Which is not only a contradiction in terms but also a bit of circular reasoning.

(My underline.)

(Full Wiki text: http://en.wikipedia.org/wiki/Fundamental_theorem_of_arithmetic)

An extension is not a contradiction, nor is it circular. It's no different than sometimes allowing 1 to be used as a prime number (which I don't think anyone has disputed can be done).

Are you now wanting to re-write the fundamental theorem of arithmetic as well as the standard definition of primes?

To force 1 to be normally considered a prime number, you'd also now have to force this '1 and -1 allowed as factors' extension of the fundamental theorem of arithmetic to be "normal". (And that now contradicts your original "simpler=preferred" contention.)


How about "Suspect Primes" to remind the user that any theory or assumption derived from them is suspect?

Oh my goodness! If it were so clear that exclusion of 1 from the set of primes would make any theorem that uses that set "suspect", why on Earth do you think 1 would have been excluded in the first place? Do you think mathemeticians just decided not to "like" 1 and threw it out of the set?


The only thing that I see from that thread as useful to this thread is to point out that the post that triggered this exposition; "one is not a prime", might have been more accurate had it said "one is not a conventional prime".

It should be quite clear by now that the standard defintion of prime number in current use excludes 1. Thus it was a fair comment. Instead of trying to argue that 1 is a prime and the standard definition is wrong, you could simply have noted that you prefer to use the older defintion.

Post #92 sums it up very well.

[aastrotech]

If it's just made up it's just made up.From Penn and Teller on astrology.

[robross]

From Penn and Teller on astrology.

The ridiculousness of this thread has just hit me. We're being trolled, of course. But the subject of the troll is the claim that 1 is a prime number. Wow.

Where else on the internet are you gonna get trolled on THAT subject?? :lol:

Rob

[Paul Beardsley]

I was wondering when someone would mention this. As it was a claim in the "marking the Moon thread" it had made me think of the monoliths in "2001 a space odyssey". I assumed that was the inspiration of the idea posted. According to the Wikipedia, the 1 4 9 ratio of the monoliths is the squares of the first three "integers".

This morning I dug out my copy of the book, as I wanted to see if back when it was written Arthur C. Clarke had used "integers" or "Primes" - and in it (my printing by Arrow, dated 1978 (I think)) he does use "integer".

I too thought of 2001. But if 1, 2 and 3 are the first three integers, what about 0, 1 and 2? Or -1, 0 and 1? Or -2003, -2002 and -2001? Or... (You get the idea/)

I believe he should have said "natural numbers". There's scope for a quibble there (is 0 a natural number?) but not as much of one.

[Paul Beardsley]

The ridiculousness of this thread has just hit me. We're being trolled, of course. But the subject of the troll is the claim that 1 is a prime number. Wow.
Giving the OP the benefit of the doubt, this thread belongs in ATM, surely?

[HenrikOlsen]

I believe he should have said "natural numbers". There's scope for a quibble there (is 0 a natural number?) but not as much of one.
Actually there seems to be more scope for a quibble than for the primes case, as natural numbers including 0 aka ℕ0 is used about as often as natural numbers without 0 aka ℕ*, so there's no real convention about which set is meant by "natural numbers".

[Paul Beardsley]

Actually there seems to be more scope for a quibble than for the primes case, as natural numbers including 0 aka ℕ0 is used about as often as natural numbers without 0 aka ℕ*, so there's no real convention about which set is meant by "natural numbers".

Indeed, although of course it would be less of a problem for monolith builders as you can't build a monolith that is 0 by 1 by 4. (I was going to put an exclamation mark* at the end of the sentence but realised it could be confused with 4-factorial.)

Incidentally, I recall from the book that the monolith was not merely 3-dimensional, so the sequence continued.

*Or "slipbanger" as they are called in Australia.

[Jeff Root]

It is clear to me that the number 'one' is a prime number (since it is
not the product of other integers), but for convenience in certain cases
in number theory, an ad hoc addition was made to the definition to
specify that it is 'not a prime'. By nature, it is prime. By definition,
it is not prime.

-- Jeff, in Minneapolis

[Merkin Muffley]

By nature, it is prime. By definition,
it is not prime.

I would put that as, by one definition, it is prime, by another definition (the one that most people use), it is not prime. I don't see one definition as more natural than the other.

[pzkpfw]

It is clear to me that the number 'one' is a prime number (since it is not the product of other integers) ...

a) Is 0.839 the product of other integers?
No.
So it's prime?
No, because by "other" you imply that the number in question is an integer.

b) So is 5 a product of other integers?
Yes. (1 and 5, and 1 isn't 5, so it's an "other" integer.)
So it's not prime?
No, because you of course expect us to already know the "1 and itself" part of the usual defintion, so "other" does not exclude 1 or itself.

c) So is -5 a product of other integers?
No (using b) ). (Just 1 and itself.)
So is -5 a prime?
No (not usually*); you possibly meant "natural numbers" not "integers".

Now, I'm not meaning to critique your short-hand definition. I know of course you didn't intend that to be a complete or fully accurate description, and I'm not playing semantic games (though possibly showing that semantic quibbling is counter-productive).

My point here is simply that something in maths needs to be consistently understood so that all mathemeticians and ordinary folk like me can all understand the same thing from them.


..., but for convenience in certain cases in number theory, an ad hoc addition was made to the definition to specify that it is 'not a prime'. By nature, it is prime. By definition, it is not prime.

-- Jeff, in Minneapolis

Primes are a kind of number. The definition of the set of those numbers must accurately describe that set. The set does not exist because of the definition. We counted 1, 2, 3, ... long before we called them the set of "natural numbers". Currently (yes, opinion has changed) 1 is not a prime, thus the definition must exclude them.

How is 1 a prime by "nature"? By what standard is that judged? (The contention of the OP of this thread was that a definition that is "simpler" is "preferred". My counter the whole time has been that a "simpler" definition is not preferred if it is incorrect; and that besides - exclusion of 1 as a prime made more subsequent defintions "simpler" than it made more "complex", so even by the claim in the OP it is better that 1 not normally be considered prime.)

You say "for convenience" and "ad hoc addition". I would say "correction", "refinement" and "improvement".


---
A note on negative primes: http://primes.utm.edu/notes/faq/negative_primes.html

[Jens]

It is clear to me that the number 'one' is a prime number (since it is
not the product of other integers), but for convenience in certain cases
in number theory, an ad hoc addition was made to the definition to
specify that it is 'not a prime'. By nature, it is prime. By definition,
it is not prime.


But I think there are two reasonable ways to define a prime. One is a number that can only be divided by 1 and itself. The other definition could be a number that can be divided by exactly two numbers, 1 and itself. By the first definition, 1 is in, in the second definition, it is out.

[Jeff Root]

Very nice reply. You didn't tear down my argument, but I think you
showed it to be weak. I can't tear down your argument, but I think
I can show it to be weak.


It is clear to me that the number 'one' is a prime number
(since it is not the product of other integers) ...
a) Is 0.839 the product of other integers?
No.
So it's prime?
No, because by "other" you imply that the number in question is an integer.
I was alluding to my post #82 in which I gave my definition:

A prime number is a positive integer which is not the product of
other integers.
I'm actually sufficiently ignorant of the subject not to know how
factoring of negative numbers is handled. If you can believe such
ignorance is possible. But it appears to be safely ignorable as long
as we stick strictly to the question of whether the number 'one' is
prime or not. So I recognize that it appears to be necessary to
restrict primes to positive numbers, but can't say exactly why.
My ignorance about this is obviously germaine to my argument, but
I hope it doesn't get in the way of my argument.

It seems obvious that my definition of prime numbers captures the
intended idea of what prime numbers are. If modern mathematics
chooses to tweak that definition, for whatever reason, it moves the
definition away from the intent. By eliminating 'one' from the set of
prime numbers, the modern definition deviates from the fundamental
concept of what a prime number is.


b) So is 5 a product of other integers?
Yes. (1 and 5, and 1 isn't 5, so it's an "other" integer.)
So it's not prime?
No, because you of course expect us to already know the "1 and itself"
part of the usual defintion, so "other" does not exclude 1 or itself.
My definition bypassed the languge about "1 and itself", by design.
I consider that language superfluous. I think it was put there as a
clarification, rather than as an essential part of the definition.

When I say "other integers" I mean precisely "other integers".
Five is not a product of other integers.


c) So is -5 a product of other integers?
No (using b) ). (Just 1 and itself.)
So is -5 a prime?
No (not usually*); you possibly meant "natural numbers" not "integers".
If the discussion can be limited to natural numbers (positive integers,
as I said in my definition), then I will be much more comfortable in my
argument. Thank you! (This relates to my question earlier in the
thread as to whether prime numbers or negative numbers were the
first to be invented/discovered, and is suggested by a comment in
the link you provided.) I meant "integers", trying to be as inclusive
as possible. I think that "integers" works. Five is not the product
of any other integers, for example, including negative integers. But
you were responding to my shorthand, as you say, not my definition
which limited primes to positive integers.


Now, I'm not meaning to critique your short-hand definition. I know
of course you didn't intend that to be a complete or fully accurate
description, and I'm not playing semantic games (though possibly
showing that semantic quibbling is counter-productive).

My point here is simply that something in maths needs to be consistently
understood so that all mathemeticians and ordinary folk like me can all
understand the same thing from them.
No argument. :D


..., but for convenience in certain cases in number theory, an ad hoc
addition was made to the definition to specify that it is 'not a prime'.
By nature, it is prime. By definition, it is not prime.
Primes are a kind of number. The definition of the set of those numbers
must accurately describe that set. The set does not exist because of
the definition. We counted 1, 2, 3, ... long before we called them the set
of "natural numbers". Currently (yes, opinion has changed) 1 is not a
prime, thus the definition must exclude them.
This really sounds like you are making my argument for me. :D


How is 1 a prime by "nature"? By what standard is that judged?
I think that my definition, that a prime number is a positive integer
which is not the product of other integers, captures the intended
meaning of "prime number". This definition, primitive and not the
currently universally-used definition, includes 'one' as a prime.


(The contention of the OP of this thread was that a definition that
is "simpler" is "preferred". My counter the whole time has been that a
"simpler" definition is not preferred if it is incorrect; and that besides -
exclusion of 1 as a prime made more subsequent defintions "simpler"
than it made more "complex", so even by the claim in the OP it is
better that 1 not normally be considered prime.)
I suspect that the orginal poster used the idea of simplicity as the
closest idea to what he meant. I used the idea of naturalness.
Both are very subjective. Okay-- mine is horribly subjective. :D
I would not interpret his preference for simplicity too literally.

Which definition is "correct" in this case just means which definition
has been accepted by convention. So being correct doesn't mean
that it is the best definition or the one that people naturally came
up with when they started thinking about relationships between
different numbers.

If you are right that the convention of excluding '1' as prime made
more subsequent defintions "simpler" than it made more "complex",
then that shows the convention to have some utility. Perhaps
enough utility to justify it.


You say "for convenience" and "ad hoc addition". I would say
"correction", "refinement" and "improvement".
It departs from the essential meaning of what a prime number is.
It adds an artificial restriction to make certain theorems simpler.
'One' clearly would be considered the quintessential prime number
if not for the desire to simplify those theorems. 'One' is a positive
integer which is not the product of other integers. It is the prime
prime number in every way except by the modern definition.

-- Jeff, in Minneapolis

[Jeff Root]

It is clear to me that the number 'one' is a prime number (since it is
not the product of other integers), but for convenience in certain cases
in number theory, an ad hoc addition was made to the definition to
specify that it is 'not a prime'. By nature, it is prime. By definition,
it is not prime.
But I think there are two reasonable ways to define a prime. One is a
number that can only be divided by 1 and itself. The other definition
could be a number that can be divided by exactly two numbers, 1 and
itself. By the first definition, 1 is in, in the second definition, it is out.
To me it is obvious that the natural definition is "A positive integer
which cannot be evenly divided by any other integers."

Following your argument, a "second" would be "A number that can
be divided by exactly three numbers." A 'tertiary" must be "A number
that can be divided by exactly four numbers." And so forth.

The essential idea of "prime numbers" is not that they ARE divisible
by a limited set of numbers, but that they are NOT divisible into
other integers. It is their indivisibility which makes them prime.

-- Jeff, in Minneapolis

[pzkpfw]

I think that my definition, that a prime number is a positive integer which is not the product of other integers, captures the intended meaning of "prime number". This definition, primitive and not the currently universally-used definition, includes 'one' as a prime.

Part of why the currently used definition of prime numbers excludes 1 is that it is even "deeper" than the 'the intended meaning of "prime number"'. 1 and -1 are "units" in the set of integers. You could argue that being a unit shouldn't preclude it also being called a prime, but that would still ignore the other reasons not to consider 1 a prime.


It departs from the essential meaning of what a prime number is.
It adds an artificial restriction to make certain theorems simpler. 'One' clearly would be considered the quintessential prime number if not for the desire to simplify those theorems. 'One' is a positive integer which is not the product of other integers.

I'm still not convinced by your "essential meaning" of what a prime is. That a prime number can only be produced by multiplying 1 and the number itself is a very specific thing. Multiplying 1 by itself seems very much the odd or trivial case; the exception.

That 5, 7 or 243112609-1 are prime is quite different to "1 is a prime".


It is the prime prime number in every way except by the modern definition.

...it's a "unit". (insert smiley)

[Jeff Root]

Part of why the currently used definition of prime numbers excludes
1 is that it is even "deeper" than the 'the intended meaning of
"prime number"'.
If the pronoun 'it' refers to "the currently used definition", then you
are saying that the definition actually defines something "deeper"
than prime numbers, rather than prime numbers. And I agree.


1 and -1 are "units" in the set of integers. You could argue that being
a unit shouldn't preclude it also being called a prime, but that would
still ignore the other reasons not to consider 1 a prime.
Nobody has given any reason for a unit not to be a prime, so the
"other reasons" are the only reasons. And those other reasons are
apparently that number theorists consider it to make some of their
theorems simpler.


I'm still not convinced by your "essential meaning" of what a prime is.
That a prime number can only be produced by multiplying 1 and the
number itself is a very specific thing.
Multiplying a number by 1 doesn't produce anything! It leaves the
number unchanged! It is an instruction to do nothing!


Multiplying 1 by itself seems very much the odd or trivial case; the
exception.
Yes. Multiplying any number by 1 is trivial. It is a non-operation.


That 5, 7 or 243112609-1 are prime is quite different to "1 is a prime".
The number one is super-prime. It is more prime than any other
integer. Its primacy among numbers is matched by its indivisibility,
which makes it the first prime number.


It is the prime prime number in every way except by the modern definition.
...it's a "unit". (insert smiley)
Yes, it is a unit. So? :D

-- Jeff, in Minneapolis

[pzkpfw]

Multiplying a number by 1 doesn't produce anything! It leaves the
number unchanged! It is an instruction to do nothing!

...

Yes. Multiplying any number by 1 is trivial. It is a non-operation.

But multiplying 1 by another number isn't.

If you have five apples and I say, just keep them - that's like a multiply by one, essentially "do nothing". But if instead you have one apple and I say take 5 as many; then you've multiplied that 1 by 5. (And you'll find that 5 can't be factored into anything but 1 and 5 - it's prime).

If you have one apple and I say, just keep it - that's like a multiply by one, essentially "do nothing". But if instead you have... um... welll... the reverse case is just the same. Again a trivial nothing. (And yes you'll find that 1 can't be factored into anything but 1 and, ... well, 1. It's not the same case.).

So I stand by the comment that 1 as a prime is quite different to all the other primes, and that the defintion of 1 as "unit" and not "prime" has a specific relevance in maths.

[Paul Beardsley]

Jeff, you seem to be thinking of primes in terms of what they are not. Think of them instead as building blocks of other numbers.

As has already been explained, primes "make" other numbers by raising them to integer powers.

For instance, 2 makes 1, 2, 4, 8, 16, 32 and so on by raising it to the power of 0, 1, 2, 3, 4, 5 and so on respectively. 3 makes 1, 3, 9, 27 and so on. 5 makes 1, 5, 25, 125 and so on. And every number that isn't a prime is a product of one or more of these factors.

Whereas 1 makes 1 and nothing else, no matter what positive power you raise it to. In other words, it's the number that doesn't do the building that is required of primes.

1 is often described as "neither prime nor composite".

I liked an earlier post which likened prime numbers to the periodic table of elements. Bear in mind that "elements" used to mean earth, air, fire and water. Now it doesn't, and water does not appear on the periodic table of elements. There may well have been people moaning about the wrongness of water being excluded - "Come on, water is the element! It's the basis of life and seven tenths of the world's surface is covered in it! What could be more elemental than that? Your exclusion of it is just semantic."

But there's no benefit to undoing progress.

[Jeff Root]

Multiplying a number by 1 doesn't produce anything! It leaves the
number unchanged! It is an instruction to do nothing! ...
Multiplying any number by 1 is trivial. It is a non-operation.
But multiplying 1 by another number isn't.

If you have five apples and I say, just keep them - that's like a
multiply by one, essentially "do nothing". But if instead you have
one apple and I say take 5 as many; then you've multiplied that
1 by 5.
Multiplying by 5 increases the number of apples. But it does not
produce a number. Multiplying any number by 1 does not produce
a number, and multiplying the number '1' by any number does not
produce a number. Multiplying the number '1' by the number '5'
results in the number '5', which is one of the two numbers you
started with, so nothing has been produced. You cannot produce
the number '5' or any other prime number by multiplying two integers
together. That is the quality that makes them prime.


So I stand by the comment that 1 as a prime is quite different to
all the other primes, and that the defintion of 1 as "unit" and not
"prime" has a specific relevance in maths.
I question that the status of the number '1' as a 'unit' has any
relevance to its status as a prime. Okay, '1' is a 'unit'. What does
that have to do with it being or not being a prime?

-- Jeff, in Minneapolis

[Paul Beardsley]

Multiplying by 5 increases the number of apples. But it does not produce a number.
Er, yes it does, Jeff.

Multiplying any number by 1 does not produce a number, and multiplying the number '1' by any number does not produce a number.
Yes it does.

Not sure what you're trying to argue here.

[HenrikOlsen]

This whole discussion is yet another example of people confusing the simplified explanation they where taught in school1 to introduce them to the subject, with understanding the subject in depth.

The set of units is the generator of all numbers by addition, the set of primes is the generator of all numbers by multiplication.

This is a deeper definition of primes that steps away from looking at natural numbers or integers and instead looks at the deeper concepts behind what numbers, addition and multiplication "really" is.

In this definition, 1 isn't prime.

1) I'm partial to the lies-to-children description:)

[Ivan Viehoff]

It is clear to me that the number 'one' is a prime number (since it is
not the product of other integers), but for convenience in certain cases
in number theory, an ad hoc addition was made to the definition to
specify that it is 'not a prime'. By nature, it is prime. By definition,
it is not prime.
I prescribe reading a book on, or taking a course in, basic factorisation theory in rings. You'll soon have a very strong sense that 1 isn't prime.

[kleindoofy]

This whole discussion is yet another example of people confusing the simplified explanation they where taught in school to introduce them to the subject, with understanding the subject in depth ...
That is a concept which can be applied to almost any subject.

The math, chemistry, physics, history, language, etc. majors I knew at the university all said they had learn how to abandon the 'facts' they had been taught at school, the recitation of which would invariably earn them a weary look from their professor.

We always got a kick out of listening to the theologians talk about Hebrew grammar. After spending years doing comparative Semitic linguistics, we knew that what we called 'theologian grammar' was sooooo off base. It was overly simplified and had answers for everything, even if there weren't any.

[Jeff Root]

Multiplying by 5 increases the number of apples. But it does not
produce a number.
Er, yes it does, Jeff.
Multiplying positive integers together produces composite numbers.
No composite number is produced by multiplying 1 and 5 together.

-- Jeff, in Minneapolis

[pzkpfw]

You cannot produce the number '5' or any other prime number by multiplying two integers together.

No.

1 x 5 = 5

1 and 5 are both integers.


Multiplying positive integers together produces composite numbers.

Not always.

1 x 5 = 5

1 and 5 are both integers, 5 is not a composite number.


No composite number is produced by multiplying 1 and 5 together.

True.

But 5 is a number. Not a composite number, but it is a number.

[Jeff Root]

You cannot produce the number '5' or any other prime number by
multiplying two integers together.
No.

1 x 5 = 5

1 and 5 are both integers.
No number was produced. You just kept the number you already had.


Multiplying positive integers together produces composite numbers.
Not always.

1 x 5 = 5

1 and 5 are both integers, 5 is not a composite number.
Again, no number was produced.


No composite number is produced by multiplying 1 and 5 together.
True.

But 5 is a number. Not a composite number, but it is a number.
Yes, but it isn't produced by the multiplication. It is a number that
you already had and knew before the multiplication.

-- Jeff, in Minneapolis

[pzkpfw]

If you have 1 apple and I multiply that by 5 for you, you end up with 5 apples.

Why would it matter you knew I had 5 (or more) apples to use?

Do we need a new thread to argue what multiplication is?

[SeanF]

Indeed, although of course it would be less of a problem for monolith builders as you can't build a monolith that is 0 by 1 by 4. (I was going to put an exclamation mark* at the end of the sentence but realised it could be confused with 4-factorial.)

Incidentally, I recall from the book that the monolith was not merely 3-dimensional, so the sequence continued.
Well, then, clearly they started with zero, it's just that the monolith's zero-length dimension is not in one of our three. :whistle:

BTW, I understand what Jeff is trying to say about "producing" numbers.

2 x 5 = 10.
1 x 5 = 5.

In the first case, you started with 2 and 5, and you produced 10. 10 is something new.

In the second case, you started with 1 and 5. Thus, it makes no sense to say you "produced" 5 - you already had it, it's not something new.

Not necessarily saying I agree with him on the relevance of it, but I get his point. :)