By MUHAMMAD YUSRAN BASRI MATH ICP B 2012

Many early writers felt that the numbers of the form 2 n -1 were prime for all primes n, but in 1536 Hudalricus Regius showed that = 2047 was not prime (it is ). By 1603 Pietro Cataldi had correctly verified that and were both prime, but then incorrectly stated 2 n -1 was also prime for 23, 29, 31 and 37. In 1640 Fermat showed Cataldi was wrong about 23 and 37; then Euler in 1738 showed Cataldi was also wrong about 29. Sometime later Euler showed Cataldi's assertion about 31 was correct.Pietro CataldiFermatEulerSometime later

Enter French monk Marin Mersenne ( ). Mersenne stated in the preface to his Cogitata Physica-Mathematica (1644) that the numbers 2 n -1 were prime for n = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127 and 257 and were composite for all other positive integers n < 257. Mersenne's (incorrect) conjecture fared only slightly better than Regius', but still got his name attached to these numbers.Marin Mersenne

Mersenne (1588 – 1648) found that 2 p -1 is prime number for p = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, dan 257. Although proved that is incorrect, but form 2 p -1 (known as mersenne numbers) still interest or made a splash. Lucas – Lehmer Lucas found condition (1870) and lehmer tested it at Tested Lucas – Lehmer : for odd number p, Mersenne numbers 2 p -1 is prime number if and only if 2 p −1 |S( p −1) with S(n+1) = (S(n)) 2 – 2 dan S(1) = 4.

Definition: When 2 n -1 is prime it is said to be a Mersenne prime.

The highest mersenne numbers

Theorem Let p be an odd prime and let q be a prime divisor of M p. Then q = 2kp + 1 for some positive integer k. Proof: From the congruence 2 p ≡ 1 (mod q) and from the fact that p is a prime, by Proposition 1.30, it follows that p is the least positive integer satisfying this property. By using Fermat’s little theorem, we have 2 q−1 ≡ 1 (mod q), hence p | (q−1), by Proposition 1.30 again. But q−1 is an even integer, so q−1 = 2kp and the conclusion follows.

Proposition A positive integer x is such that a x ≡ 1 (mod m) if and only if x is a multiple of the order of a modulo m.

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