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?

(In reply to A bet by Federico Kereki)

Aha -- that is the missing piece for my proof idea.

Suppose there was a sequence of primes with first term T>2. Of course the first term mod T is 0, but if any other term mod T is 0, the term wouldn't be prime.

However, a chain of numbers will form, but since there are fewer that T numbers to choose from, a 0 will eventually will repeat, and thus the number isn't prime. (Note that T-1 and only T-1 will chain to itself, as 2(T-1)+1 = 1*T+(T-1))

(Also note that for T=2, the sequence 2, 5, 11, 23... the sequence to be checked should be 5, 11, 23...)

Posted by Gamer on 2006-11-29 20:35:27