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

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.)

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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.

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

Your OP was clearly about whether the set of p^rim_{e n}u^mbe_rs (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:

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.

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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.

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.

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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.

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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.

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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!

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"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.

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1=2^0, case closed

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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, or this wiki link, for example.

Nick

edit: apparently the value of 0⁰ is contentious.

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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.

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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.

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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

It may be helpful to see where this thread was spun off from: Marking the moon, a message to the future.

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.

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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.

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?

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

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.

Which were invented first? Prime numbers or negative integers?

-- Jeff, in Minneapolis

Prime numbers (probably), but we could argue as to whether they were invented or discovered.

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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.

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.

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.

You keep trying to tread water by grabbing at words.

Your old idea is dragging you down.

Glub.

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0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 ...

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