The Lucas-Lehmer test is an efficient deterministic primality test for determining if a Mersenne number  is prime. Since it is known that Mersenne numbers can only be prime for prime subscripts, attention can be restricted to Mersenne numbers of the form , where  is an odd prime.

Consider the recurrence equation

(1)

with . For example, ignoring the congruence, the first few terms of this iteration are 4, 14, 194, 37634, 1416317954, ... (OEIS A003010).

It turns out that  is prime iff , and the value  is called the Lucas-Lehmer residue for .

For example, the sequence obtained for  is given by 4, 14, 67, 42, 111, 0, so  is prime.

For prime , the first few Lucas-Lehmer residues are 1, 0, 0, 0, 1736, 0, 0, 0, 6107895, 458738443, 0, 117093979072, ... (OEIS A095847).

This test can also be extended to arbitrary integers. Prior to the work of Pratt (1975), the Lucas-Lehmer test had been regarded purely as a heuristic that worked a lot of the time (Knuth 1969). Pratt (1975) showed that Lehmer's primality heuristic could be made a nondeterministic procedure by applying it recursively to the factors of , resulting in a certification of primality that has come to be known as the Pratt certificate.

A generalized version of the Lucas-Lehmer test lets

(2)

with  the distinct prime factors, and  their respective powers. If there exists a Lucas sequence  such that

(3)

for , ...,  and

(4)

then  is a prime. This reduces to the conventional Lucas-Lehmer test for Mersenne numbers.

REFERENCES:

Knuth, D. E. §4.5.4 in The Art of Computer Programming, Vol. 2: Seminumerical Algorithms. Reading, MA: Addison-Wesley, 1969.

Pratt, V. "Every Prime Has a Succinct Certificate." SIAM J. Comput. 4, 214-220, 1975.

Ribenboim, P. "Primality Tests Based on Lucas Sequences." §2.V in The Little Book of Bigger Primes, 2nd ed. New York: Springer-Verlag, p. 63, 2004.

Sloane, N. J. A. Sequences A003010/M3494 and A095847 in "The On-Line Encyclopedia of Integer Sequences."