Pick a positive integer to start a sequence. Now double it, and add one to the result: this is the second number of your sequence. Double that number, and add one, and that will be your third number; repeat the doubling and adding, and you will have a fourth number, and so on.

If you start with a prime number, and you keep doubling and adding one, is it possible to produce a sequence with only prime numbers?

let p, 2p+1, ... be a sequence with p an odd prime. The kth term is  2^(k-1)*p + 2^(k-1) - 1. So the pth term is

 2^(p-1)*p + 2^(p-1) - 1. By Fermat's thm. 2^(p-1)-1 = 0 mod p so p divides the entire pth term.

 Of course in the case beginning with 2, 5, ...   the sixth term is divisible by 5.

Posted by Dennis on 2006-11-30 09:41:36