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 start the sequence.  Then the elements of the sequence are N_k=(2^k)*(P+1)-1, k=0,1,2,... . When  P+1 does not end in the digit 0, there is 2^k times it that ends in 6 so that N_k is divisible by 5. When P+1 ends in 0 but is not divisible by 3, then there is 2^k times it that is congruent to 1 mod 3, so that N_k is divisble by 3. This leaves the P's such that P+1 is divisble by 30, but there is 2^k times such a P+1 that is congruent to 1 mod 7.

Posted by Richard on 2006-11-29 12:17:45