Define a Prime
 

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=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]

non-negative integers that are only divisible by themselves and the units in the ring of integers

An integer so that any product equal to the integer must involve the integer itself, units, and nothing else.

Alex

[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]

[spoiler]Let (p,q) be non-negative integers such that N=p*q and p>q

If for all (p,q) p=N, then N is prime.
[/spoiler]

Pau

Here's another try:

[spoiler]A positive integer p is prime if and only if the only solutions to the equations p = a*b = c*d occur when a is not equal to b and a = c and b = d or a = d and b = c. (I almost said that the equation p = a*b has exactly two solutions (a,b) where a and b are positive integers, but then realized that I was violating the terms of the problem!)[/spoiler]

A prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors, which are (4 - 3) and the prime number itself.
:grin: Hooray for loopholes! :grin:

[QUOTE=Mini-Geek;99329]that has exactly [B]two[/B] [/QUOTE]

Rule violation. Five yards penalty.

Alex

[QUOTE=TravisT;99307]non-negative integers that are only divisible by themselves and the units in the ring of integers[/QUOTE]

Counter-example.

3 is a non-negative integer

3 is divisible by itself --- meets your definition

3 is divisible by 1, a unit in the ring of integers --- meets your definition

3 is divisible by -1, a unit in the ring of integers --- meets your definition

-3 is an element of the ring of integers

3 is divisible by -3


Therefore, according to your definition, 3 is not a prime.



Paul

Same problem with my definition. Let's try

An integer so that any product equal to the integer must involve the integer itself or one of its associates, any number of units, and nothing else.

Alex

[quote=akruppa;99330]Rule violation. Five yards penalty.

Alex[/quote]Ok, fine, here's the updated version:

A prime number (or a prime) is a natural number that has exactly (5 - 3) (distinct) natural number divisors, which are (4 - 3) and the prime number itself.
:grin: Hooray for loopholes! :grin:

One may try to invent something analytical from
[url]http://mathworld.wolfram.com/LandausFormula.html[/url]

:-)