So, do negative prime numbers exist?

[NikitaUtiu]

I know that the fundamental theorem of arithmetic states that any integer greater than 1 can be written as an unique prime factorization.
I was wondering if there is any concept of negative prime numbers, because any integer greater than 1 or less than -1 should be able to be written as n = p1 ^ e1 * p2 ^ e2 ... or - (p1 ^ e1 * p2 ^ e2 ...).

Thanks, NikitaUtiu!

 

[Goongyae]

I understand what you are saying. But the problem with admitting -1 (or 1) as a prime number is that the fundamental theorem of arithmetic wouldn't work any more. The representation of numbers in terms of primes would be ambiguous. For example, 6 = 2x3 = 2x3x(-1)x(-1). A lot of the theorems on prime numbers would need to be made more complicated to deal with -1.

For this reason, the units http://en.wikipedia.org/wiki/Unit_(ring_theory ) 1 and -1 are usually treated as separate from prime numbers.

Another idea to consider is to admit negative powers of primes. For example 5/2 is unambiguously expressible as 5^1 x 2^-1. You can use this idea to help define values for the von Mangoldt http://en.wikipedia.org/wiki/Von_Mangoldt_function and Chebyshev functions at argument 0<x<1. At x = 1 there is a discontinuity owing to the fact that 1 is a power (specifically, the zeroth power) of every prime; the magnitude of the discontinuity is log(4pi^2)=log(2x3x5x7x11x...). When you add this log(4pi^2) to the so-called "explicit formula" (http://en.wikipedia.org/wiki/Explicit_formula), you see there is now a log(2pi) for x>1 and a -log(2pi) for x < 1. Extending these functions in this way is an interesting concept but isn't all that useful AFAIK.

You might also be interested in learing about the Gaussian integers http://en.wikipedia.org/wiki/Gaussian_integer.
Basically a+bi where a and b are integers can be factored into "Gaussian primes" . Things are more complicated there, with more units and more kinds of primes.

 

[mathwonk]

the theorem says that every integer which is neither zero nor invertible, i.e. not 0, 1, nor -1, can be written as a product of prime integers,. i.e. integers which are themselves neither 0,1, nor -1, and also that have as factors only themselves, minus themselves,a nd 1, -1.

the factorization is not unique, but in any two factorizations, the absolute values of the factors are the same.for example the integer -12 is the product of the primes (-3)(2)(2), and also of (3)(-2)(2), and also (-3)(-2)(-2).