ntt@dciem.UUCP (Mark Brader) (05/09/84)
While checking up the details about Carmichael numbers for the message I just posted about the number 1729, I noticed that the same article (by Carl Pomerance in the December 1982 Scientific American) contained a list of all the Mersenne primes known at that time. Gardner's article referred to in my query must have been written a little later than I thought; the 24th Mersenne prime, referred to in it, was discovered in 1971. Well, there have been just 3 Mersenne primes discovered from 1971 to 1982: M[21701], M[23209], and M[44497], where M[n] means 2^n -1, of course. 01, 09, and 97 are all congruent to 1 modulo 4, so the sequence of last digits of perfect numbers continues with three more 6's. Wouldn't it be interesting if all further ones did also? I may as well give the complete list of last digits of perfect numbers again: 6 8, 6 8; 6 6 8 8, 6 6 8 8; 6 8 8 8, 6 6 6 8; 6 6 6 6 6 6 6. Anyone know if any more Mersenne primes (and hence perfect numbers) have been discovered since 1982? The article mentioned that numbers up to M[62982] had been searched ... notice, 26 Mersenne primes in the first 24,000 possibilities, and then just one in the next 38,000. Of course, whether the number of Mersenne primes is finite or infinite is unsolved. Mark Brader