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
One Approach (Spoiler?) by Richard)
I get
N_k = 2^(k-1) * (P+1) - 1
as the non-recursive formula.
According to Fermat's Little Theorem, there exists a (k-1) such that
2^(k-1) = m * P + 1
where m is some non-negative integer. So,
N_k = [m*P + (m+1)] * P
Which is not prime. So the answer is: no, it is not possible to produce a purely prime sequence using this method.
re: One Approach (Spoiler?)
