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

What are the rules that can affirm this?

Thanks in advance and happy new year to all.

Best regards,

  • $\begingroup$ Any integer with no proper divisors is prime (ex. -3) $\endgroup$ – user261263 Jan 14 '18 at 16:57
  • $\begingroup$ This is a very near duplicate of another question which got lot's of attention - see the link. $\endgroup$ – Mark Bennet Jan 14 '18 at 16:57
  • $\begingroup$ Of course there are negative prime numbers. But do take care to define the prime counting function so that $\pi(x) = \pi(-x)$. $\endgroup$ – Robert Soupe Jan 15 '18 at 8:02

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.


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