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# examples of the Lucas-Lehmer primality test on small numbers

To illustrate the Lucas-Lehmer primality test for Mersenne numbers, it is best to use small numbers. For any purpose other than pedagogical, applying this test to a small number is of course more trouble than it’s worth, since it would be much easier to just perform integer factorization; a classic example of “sandblasting a soup cracker,” in Scott Adams’ words. But for the purpose of illustrating the test, since the numbers in the recurrence relation get quite large, small numbers are more suitable.

Is $M_{3}=2^{3}-1=7$ a Mersenne prime? The first few terms of the recurrence relation $s_{n}={s_{{n-1}}}^{2}-2$ (with initial term $s_{1}=4$) are 4, 14, 194, 37634, 1416317954, 2005956546822746114, 4023861667741036022825635656102100994, etc. (see A003010 in Sloane’s OEIS). For our first example, we need to look up $s_{{3-1}}$, which is 14. That number divided by 7 is 2, telling us that 7 is indeed a Mersenne prime.

Is $M_{7}=2^{7}-1=127$ a Mersenne prime? $s_{6}=2005956546822746114$ and that divides neatly by 127, being 15794933439549182 times 127. So 127 is indeed a Mersenne prime. Already this is larger than what a typical pocket scientific calculator can handle with integer precision.

What about 11? $s_{{10}}$ is 687296824066442772388374862317475309242471541086466717521926185830884874057909 579647328830691025610434367796639355951720423573065949163446060745647128680782 876080552030246583594390175808839109786661858757174155410844949265004751673811 68505927378181899753839260609452265365274850901879881203714, which divided by 2047 gives a remainder of 1736, telling us that 2047 is a Mersenne number but not a Mersenne prime. Even by computer it would have been quicker to perform trial division on 2047, giving the result $2047=23\times 89$. But already for $M_{{59}}$, with its least prime factor being 179951, the 16336th prime, the Lucas-Lehmer test appears much more attractive than trial division.

How does the test work on a composite number, say 4? Seeing that $s_{3}=194\equiv 14\mod 15$ (or $-1\mod 15$ if you prefer), it is quite obvious that 15 is not a Mersenne prime.

## Mathematics Subject Classification

11A51*no label found*

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