Lucas-Lehmer Test

The Lucas-Lehmer test is an efficient deterministic primality test for determining if a Mersenne number M_n 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 M_p=2^p-1, where p is an odd prime.

Consider the recurrence equation

 s_n=s_(n-1)^2-2 (mod M_p)

with s_0=4. For example, ignoring the congruence, the first few terms of this iteration are 4, 14, 194, 37634, 1416317954, ... (OEIS A003010).

It turns out that M_p is prime iff s_(p-2)=0 (mod M_p), and the value s_(p-2) (mod M_p) is called the Lucas-Lehmer residue for p.

For example, the sequence obtained for p=7 is given by 4, 14, 67, 42, 111, 0, so M_7=127 is prime.

For prime p, 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 n-1, 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


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


for j=1, ..., n and

 U_(N+1)=0 (mod N),

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

See also

Lucas Sequence, Mersenne Number, Mersenne Prime, Pratt Certificate, Rabin-Miller Strong Pseudoprime Test

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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."

Referenced on Wolfram|Alpha

Lucas-Lehmer Test

Cite this as:

Weisstein, Eric W. "Lucas-Lehmer Test." From MathWorld--A Wolfram Web Resource.

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