[seqfan] The iterations (x -> 2x-1) and (x -> 2x+1) of primes

[Tomasz Ordowski]

Dear readers!

The recursion a(n) = 2 a(n-1) - 1 with a(0) = p is given by the formula
a(n) = (p-1)2^n+1.
The recursion b(n) = 2 b(n-1) + 1 with b(0) = p is given by the formula
b(n) = (p+1)2^n-1.

Problem:
Are there primes p such that both a(n) and b(n) are all composite for 0 < n
< p ?

Conjecture:
If p is a prime, then there exists 0 < n < p such that a(n) or b(n) is
prime.

Try to disprove this strong conjecture!

Best regards,

Thomas Ordowski
________________
Odd primes p such that both (p-1)2^n+1 and (p+1)2^n-1 are all composite for
1 < 2^n < p
are 613, 1301, 1373, 1933, 6373, 7127, 9851, 11383, 11443, 13121, 14207,
14293, 23021, ...
Data from Amiram Eldar.