Are there negative prime numbers?

Question

It seems generally admitted that there are no negative prime numbers.

What are the rules that can affirm this?

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This is a very near duplicate of another question which got lot's of attention - see the link. — Mark Bennet, Jan 14 '18 at 16:57
Of course there are negative prime numbers. But do take care to define the prime counting function so that $\pi(x) = \pi(-x)$. — Robert Soupe, Jan 15 '18 at 08:02

score 2 · Answer 1 · answered Jan 14 '18 at 16:59

This is false. $-2$ is prime. One of the two following statements (depends a bit on context) is the definition of primarily.

Indivisibility: A number $p$ is prime if it doesn’t have any factors other than itself and $1$, up to unit multiples.

Note: “up to unit multiples” allows us to ignore the fact that $-1|7$ or $i|3i$

Primality: A number $p$ is prime if whenever $p|ab$ ether $p|a$ or $p|b$.

In the integers these definitions are equivalent, but for other sets they might not be. In other sets, we call the second the definition of primality usually. However, $-7$ is a prime integer according to both of these definitions. Although lay people might claim that there aren’t any negative primes, but there’s no mathematical basis for this claim.

Are there negative prime numbers?

1 Answers1