Imagine a precisely cut plank of wood "p" inches** long. The width and height do not matter but the shape is a perfect rectangular solid. Along the length, each inch is marked with a line.

If that board can not be cut into pieces along the inch markers ONLY such that you can rearrange them laterally (and without turning the pieces) into a rectangular shape that has different dimensions than the original board (breath!) then p is prime.

** inch is arbitrary of course.

The turning limitation might not be necessary. I'd have to think about that some more.

[QUOTE=ewmayer;99515]An integer p is prime iff no integer x in the range sqrt(p) <= x < p divides p.[/QUOTE]

An integer p is prime if and only if for all A,B, such that A^2 = B^2 mod p,
then A=B mod p or A=-B mod p.

[QUOTE=R.D. Silverman;99523]An integer p is prime if and only if for all A,B, such that A^2 = B^2 mod p,
then A=B mod p or A=-B mod p.[/QUOTE]

Ah, but you had to explicitly reference the number 2 - that's a 10-yard penalty and loss of down for you, Bob. Oh wait, I just referenced 1 and 0 in my description of the penalty - that's [i]log(p)/log(sqrt(p))[/i] minutes in the penalty box for me...

Aren't arbitrary puzzle rules great?

[QUOTE=ewmayer;99526]Ah, but you had to explicitly reference the number 2 - that's a 10-yard penalty and loss of down for you, Bob. Oh wait, I just referenced 1 and 0 in my description of the penalty - that's [i]log(p)/log(sqrt(p))[/i] minutes in the penalty box for me...

Aren't arbitrary puzzle rules great?[/QUOTE]

So change it to A*A and B*B. This isn't an issue of using the
number 2, but rather one of notation.

[QUOTE=R.D. Silverman;99530]So change it to A*A and B*B. This isn't an issue of using the number 2, but rather one of notation.[/QUOTE]I'm amazed that no-one has yet picked me up for using ln(e) in a polynomial to be evaluated at integer values of its variables. Had anyone done so I would have, of course, replaced them all with (x/x)

Paul

[QUOTE=S485122;99478]Isn't insisting on the plural the same as saying "more than one" and thus an explicit reference to a specific integer ? Or would this be a to restrictive interpretation of the rules ?[/QUOTE]

Can the high command give a ruling on this? I can reword my answer to say "a person"<->"one".

[QUOTE=xilman;99541]I'm amazed that no-one has yet picked me up for using ln(e) in a polynomial to be evaluated at integer values of its variables. Had anyone done so I would have, of course, replaced them all with (x/x)

Paul[/QUOTE]

Paul,

Using x/x would necessitate x\neq 0.

Prime Number definitions:

1) A positive integer p is a prime or a prime number if it is a whole number larger than unity and its only positive divisors are the unit and itself.
Every prime number has the property that if it divides a product then it must divide at least one of the factors [Euclid c.300 BC.]

2) A positive integer is prime if only *two* but only *two* distinct factors, are itself and unity

Euclid books 7 and 8 regards a number as a line interval compounded of units and
defines a prime as a number which can only be measured by the unit (not itself a number)
It follows from both the above two definitions that unity is not a prime

3) A prime is an irreducible element of a unique factorization domain and is known as prime.
These are for positive primes.

To allow for negative primes we cannot define it without naming some numbers.


Definition: the term can also be used in some other situations where division is meaningful.
For instance in the context of all the integers an integer other than 0+-1 is a prime integer if its only integer divisions are +- 1 and +- n.
Mally.

[QUOTE=mfgoode;99600]Prime Number definitions:[/QUOTE]Mally,

I like to believe that all participants to the thread so far are well aware of the definition of a prime number. But you overlooked the purpose of this thread. The original question was :
[QUOTE=davar55;99304]Can you define prime numbers over the non-negative integers without any explicit reference to 0 or 1 or 2 or any other specific integer?[/QUOTE]

[QUOTE=Zeta-Flux;99599]Paul,

Using x/x would necessitate x\neq 0.[/QUOTE]Correct, and I should have stated that fact in the form "for values of x where x/x is defined".

Thanks for drawing it to my attention.

The purpose behind the formulation of this admittedly kind of
arbitrary question was to try to come up with an alternate,
nonarbitrary explanation for why 1 isn't prime (alternate to
the fundamental theorem of arithmetic's desirable simplicity).
After all, a typical definition of primality in the natural numbers
uses 1 twice:
( n is prime iff n>1 and a|n implies a=n or a=1 )
which begs the question.

But my version of the definition probably still does too:

Among the non-negative integers,
n is prime iff ((n=ab IMPLIES (n=a OR n=b)) AND NOT (n*n=n)).

This excludes 1 (and 0) from primality by using a trivial property
(n*n=n) shared only by 1 and 0, without explicitly mentioning 1
or "unit". So by this definition, 1 is not prime. But as simple as
it may be, it's still a bit arbitrary to put this into the definition, isn't it?
Yet this is all I was getting at when posing the question.