Mersenne numbers are prime or composite. A number is called perfect if it is equal to the sum of its divisors, not including itself. As an example, 6 is a perfect number, since its only divisors are 1, 2, and 3 (besides 6) and these sum to 6. Similarly, the divisors of 28, excluding 28 itself, are 1, 2, 4, 7, and 14, and these sum to 28. These numbers were considered perfect, substantiated by the creation of the universe in 6 days and of the lunar cycle in 28 days. Over 2 000 years ago, Euclid proved that every number of the form 2P-1(2P 1), where is a prime and 2P ? 1 is also a prime, must be an even perfect number. As an example, take = 2. Since 22 ? 1 = 4 ? 1 = 3 is a prime, = 2 may be used in the formula to yield a perfect number. Sub stituting, we obtain 22_1(22 1) = 2 X 3 = 6, the first even perfect number. In the eighteenth century, Euler proved that Euclid's formula gives all even perfect numbers. The first five perfect numbers given by Euclid's formula are 6, 28, 496, 8 128, and 33 550 336. According to Euclid's formula, we are only allowed to substitute primes such that 2P 1 is also prime. Let Mp stand for the number 2P 1; thus Mz = 23 1 = 7, M5 = 2s 1 = 31, and so on. If Mp is a prime, we call that particular Mp a Mer senne prime, named after the seventeenth century mathematician Marin Mersenne. As an example, Mz = 23 1 = 7, a prime; hence 7 is a Mersenne prime. Up to the present time the only known values of the prime that yield Mersenne primes are 2,3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1 279, 2 203, 2 281, 3 217, 4 253, 4 423, 9 689, 9 941, 11 213, 19 937 (24 values). It is to be understood that the omitted primes between these do not yield Mersenne primes; for example, for = 11, Mn = 211 1 = 2 047 = 89 23. In addi tion, if is composite, it is clear that Mn is composite; in fact, 2ab 1 (a and b greater than 1) is divisible by Ia 1. In particular then, if is even and larger than 2, Mn is composite. In 1772, Euler proved that Mzl = 2 147 483 647 is a prime, and this number became the largest known prime for more than 100 years. E. V. Lucas (1877, pp. 184-239, 289-321) developed a beautiful test to determine, for a given prime /?, whether Mp is prime or not. The result was later modified and the proof simplified by D. H. Lehmer (1930, pp. 419-48). Consider the following sequence of numbers, which is called the Lucas-Lehmer sequence: 4, 14, 194, 37 634, 1 416 317 954, and so on. This sequence satisfies the recurrence relation Si = 4, Sk+1 = Sk2 2, k = 1, 2, 3,. ... Thus 14 = 42 2, 194 = 142 2, and so on. The following statement can be proved:

https://www.jstor.org/stable/pdfplus/27960511.pdf

Another Way of Finding Prime Numbers.

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Another Way of Finding Prime Numbers.