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If a number fails Miller's primality test for some base a, it is not a prime. If the number passes, it may be a prime. A composite number passing Miller's test is called a ...

A pseudoprime is a composite number that passes a test or sequence of tests that fail for most composite numbers. Unfortunately, some authors drop the "composite" ...

An odd prime p is called a cluster prime if every even positive integer less than p-2 can be written as a difference of two primes q-q^', where q,q^'<=p. The first 23 odd ...

When P and Q are integers such that D=P^2-4Q!=0, define the Lucas sequence {U_k} by U_k=(a^k-b^k)/(a-b) for k>=0, with a and b the two roots of x^2-Px+Q=0. Then define a ...

Lehmer's totient problem asks if there exist any composite numbers n such that phi(n)|(n-1), where phi(n) is the totient function? No such numbers are known. However, any ...

Let p be an odd prime and b a positive integer not divisible by p. Then for each positive odd integer 2k-1<p, let r_k be r_k=(2k-1)b (mod p) with 0<r_k<p, and let t be the ...

An odd power is a number of the form m^n for m>0 an integer and n a positive odd integer. The first few odd powers are 1, 8, 27, 32, 64, 125, 128, 216, 243, 343, 512, ... ...

A prime gap of length n is a run of n-1 consecutive composite numbers between two successive primes. Therefore, the difference between two successive primes p_k and p_(k+1) ...

The distinct prime factors of a positive integer n>=2 are defined as the omega(n) numbers p_1, ..., p_(omega(n)) in the prime factorization ...

A primality test that provides an efficient probabilistic algorithm for determining if a given number is prime. It is based on the properties of strong pseudoprimes. The ...