OFFSET
1,3
COMMENTS
This sequence uses the centered version of mod. The residue system modulo prime(n) is {-1*floor(prime(n)/2)..floor(prime(n)/2)}. This is so that this sequence will encode information about the numbers around 2^A007013(4). If a(n) = k and prime(n) < 2^A007013(4) - k, then 2^A007013(4) - k is not prime (prime(n) is a factor of 2^A007013(4) - k). For example, a(22) = -3, so prime(22) = 79 is a factor of 2^A007013(4) + 3.
The length of this sequence is the lowest value of n such that A014664(n) = A007013(4). This is because for any power of 2, 2^p, if p == 0 (mod A014664(n)), then 2^p == 1 (mod prime(n)) (prime(n) is a factor of A000225(p)). Since A007013(4) is prime, we can apply this to get: If A014664(n) = A007013(4) and prime(n) < A007013(5), then A007013(5) is not prime (prime(n) is a nontrivial factor).
For any n such that prime(n) < 5*(10^51 + 5*10^9), a(n) != 1.
LINKS
Chris K. Caldwell, Mersenne Primes: History, Theorems and Lists (5. Conjectures and Unsolved Problems).
I. S. Eum, A congruence relation of the Catalan-Mersenne numbers, Indian J Pure Appl Math, 49 (2018), 521-526.
Robert Delion, The n2 + 1 Fermat and Mersenne prime numbers conjectures are resolved, Theoretical Mathematics & Applications, vol.6, no.1, 2016, 15-37.
FORMULA
a(n) = 2^(2^127 - 1) mod prime(n).
PROG
(PARI) A353214(n)=my(CM4=shift(1, 127)-1); centerlift(Mod(2, prime(n))^CM4)
CROSSREFS
KEYWORD
sign,fini
AUTHOR
Davis Smith, Apr 30 2022
STATUS
approved