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Each prime factor p_i^(alpha_i) in an integer's prime factorization is called a primary.

A prime factorization algorithm.

Two numbers which are relatively prime.

For N=k·2^n+1 with k odd and 2^n>k, if there exists an integer a such that a^((N-1)/2)=-1 (mod N), then N is prime. A prime of this form is known as a Proth prime.

A test which always identifies prime numbers correctly, but may incorrectly identify a composite number as a prime.

A double Mersenne number is a number of the form M_(M_n)=2^(2^n-1)-1, where M_n is a Mersenne number. The first few double Mersenne numbers are 1, 7, 127, 32767, 2147483647, ...

B_(p+k)=B_k+B_(k+1) (mod p), when p is prime and B_n is a Bell number.

If a prime number divides a norm but not the bases of the norm, it is itself a norm.

A number n satisfies the Carmichael condition iff (p-1)|(n/p-1) for all prime divisors p of n. This is equivalent to the condition (p-1)|(n-1) for all prime divisors p of n.

Erdős proved that there exist at least one prime of the form 4k+1 and at least one prime of the form 4k+3 between n and 2n for all n>6.