# mpz_probab_prime_p and negative inputs

Niels Möller nisse at lysator.liu.se
Mon Mar 3 21:26:48 UTC 2014

```It seems mpz_probably_prime_p considers negated primes to also be prime.
E.g, for n == -29 it returns 2, meaning "definitely prime".
Mathematically, I think -29 is usually considered neither prime, nor
composite (its prime factorization is -1 * 29 and the "unit" factor -1
kind-of doesn't count, since *if* we count it as a "real" factor, that
breaks uniqueness of factorization: we get additional possible
factorizations like -29 = (-1)^3 29, and 29 = (-1)^2 29.

I would have expected mpz_probably_prime_p to only return non-zero for
elements in the usual set of "prime numbers": 2, 3, 5, 7, 11, ...

The behaviour is also a bit inconsistent with mpz_nextprime, which for
input -29 returns the "standard" prime 2, not -23 which is the next
larger number which is considered prime by mpz_probably_prime_p.

The result of mpz_probably_prime_p for negative inputs is not
clearly documented.

Some options:

1. Document current behaviour.

2. Explicitly document that negative inputs are not allowed.

3. Change mpz_probably_prime_p to consider all integers < 2 to be
non-primes.

I'd lean towards (3), but I don't know if anyone depends on the current
behaviour.

Regards,
/Niels

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