So, do negative prime numbers exist?

In summary, the fundamental theorem of arithmetic states that any integer greater than 1 can be written as a unique prime factorization. There is no concept of negative prime numbers because it would make the theorem ambiguous. However, the idea of negative powers of primes can be used to extend the concept of prime numbers, such as with the Gaussian integers. The factorization of an integer is
  • #1
NikitaUtiu
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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!
 
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  • #2
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 [Broken]) 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.
 
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  • #3
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).
 

What is integer prime factorization?

Integer prime factorization is the process of breaking down a positive integer into its prime factors, which are the smallest numbers that can divide evenly into the given integer.

Why is integer prime factorization important?

Integer prime factorization is important because it allows us to find the unique prime factors of a number, which can be useful in various mathematical and scientific applications such as cryptography and number theory.

How do you find the prime factors of a number?

The easiest way to find the prime factors of a number is through the use of a factor tree. Start by dividing the number by its smallest prime factor, then continue dividing each resulting factor by its smallest prime factor until all factors are prime numbers.

What is the difference between prime and composite numbers?

Prime numbers are numbers that have only two factors, 1 and itself. Composite numbers, on the other hand, have more than two factors. Prime numbers are also only divisible by 1 and itself, while composite numbers have additional factors.

Can every positive integer be expressed as the product of prime numbers?

Yes, every positive integer can be expressed as the product of prime numbers. This is known as the fundamental theorem of arithmetic, which states that every integer greater than 1 is either a prime number or can be written as a unique product of prime numbers.

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