Update Seminar Form
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 2^11-1 = 2047 was not prime (it is 23.89). By 1603 Pietro Cataldi had correctly verified that 2^17-1 and 2^19-1 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.
Enter French monk Marin Mersenne (1588-1648). 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.
: When 2^n-1 is prime (where n is also a prime) it is said to be a Mersenne prime.
It was obvious to Mersenne's peers that he could not have tested all of these numbers (in fact he admitted as much), but they could not test them either. It was not until over 100 years later, in 1750, that Euler verified the next number on Mersenne's and Regius' lists, 2^31-1, was prime. After another century, in 1876, Lucas verified 2^127-1 was also prime. Seven years later Pervouchine showed 2^61-1 was prime, so Mersenne had missed this one. In the early 1900's Powers showed that Mersenne had also missed the primes 2^89-1 and 2^107-1. Finally, by 1947 Mersenne's range, n < 258, had been completely checked and it was determined that the correct list is:
n = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107 and 127.
Are there infinitely many Mersenne primes? Equivalently we could ask: Are there infinitely many even perfect numbers? The answer is probably yes (because the harmonic series diverges).
Another question that I am personally confused with is the fact that actually in the form 2^p-1, p has to be prime or just be an odd number?
Prof. Ashay Dharwadker