Define a Prime

[grandpascorpion]

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.

[R.D. Silverman]

Originally Posted by ewmayer

An integer p is prime iff no integer x in the range sqrt(p) <= x < p divides p.

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.

[ewmayer]

Originally Posted by R.D. Silverman

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.

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 log(p)/log(sqrt(p)) minutes in the penalty box for me...

Aren't arbitrary puzzle rules great?

[R.D. Silverman]

Originally Posted by ewmayer

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 log(p)/log(sqrt(p)) minutes in the penalty box for me...

Aren't arbitrary puzzle rules great?

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

[xilman]

Originally Posted by R.D. Silverman

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

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

[Uncwilly]

Originally Posted by S485122

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 ?

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

[Zeta-Flux]

Originally Posted by xilman

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

Paul,

Using x/x would necessitate x\neq 0.

[mfgoode]

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.

[S485122]

Originally Posted by mfgoode

Prime Number definitions:

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 :

Originally Posted by davar55

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?

[xilman]

Originally Posted by Zeta-Flux

Paul,

Using x/x would necessitate x\neq 0.

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.

[davar55]

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.