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Nope. Prime numbers are only positive.helen wrote:Hopefully this will be a quick one, but does anyone know if negative numbers can be prime?
I'm thinking yes, but can someone please confirm? Thanks!
It is not because primes were defined before negatives that negatives are not considered prime. Mathematicians easily adapted other concepts (evens/odds, for example) to negative numbers. Negatives are not considered prime because if they were, the Fundamental Theorem of Arithmetic (unique factorization into primes) would no longer be true. If -2 were prime, for example, we could 'prime factorize' 4 in two different ways: as 2^2 and as (-2)^2. If only positive numbers are considered prime, then every positive integer larger than 1 has one and only one prime factorization.Stacey Koprince wrote:This can trip people up b/c the technical definition would seem to allow negative numbers but, as Eric says, primes are only positive. The reason: the concept of prime was developed before the concept of negative numbers was developed. So primes were "by definition" positive b/c negative numbers didn't "exist" yet.
The most common definition of a prime is the following: 'primes are positive integers which are divisible by precisely two distinct positive integers.' If you remove the word 'positive' from the definition, the only primes would be 1 and -1 (which both have two integer divisors, 1 and -1). 3 would not be prime- it is divisible by 1, -1, 3 and -3.gmatutor wrote:Of course for the GMAT all primes are positive.
However, negative one satisfies the Fundamental Theorem and the most common definition of a prime.
Positive one is not considered prime because it is divisible be exactly one integer rather than two.
Negative one on the other hand is divisible by exactly two integers (+1 and -1). Thus negative one satisfies the definition of a prime.
Yes, but the restriction you need to impose on the powers of -1 here are similar to the restrictions as you would need to impose on the power on 1 in a prime factorization if you permitted 1 to be prime. Mathematicians do not permit 1 to be prime for this very reason; if 1 is not prime, the Fundamental Theorem of Arithmetic can be stated without any restrictions on powers.gmatutor wrote: Since that is the case, the Fundamental Theorem is satisfied if we do not let the prime factor negative one have a power other than one. In other words we cannot factor 5 into 5 X -1^2.
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