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 I would take that bet
I only considered prime "p" values, and Fermat's theorem also has to do with primes...