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.

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:

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

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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")

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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.

Elegantly put, cjameshuff.

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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.

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.

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.

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.You do have to include that in the definition, though, right? Else what do you do with 3.5 / .5 = 7?

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.

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personal page: http://blog.astrosketches.info

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

Think modern.

Please.

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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.

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
http://mathworld.wolfram.com/PrimeNumber.html
http://primes.utm.edu/notes/faq/one.html

etc.

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As this post mentions, it's possible to do 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.

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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.

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.

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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.

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.

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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.

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'.

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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,

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Back on topic:

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

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_Subprime.

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Meet the OOONG TOE.

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.

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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.

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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.