cp's OEIS Frontend

Prime terms from A279884.

If a(4) exists, it must be bigger than (prime(2200)-1)^(prime(2200)+1) + 1 = 19422^19424 + 1.

Corresponding pairs of numbers (j, prime(j)): (1, 2); (2, 3); (11, 31); ...

It is extremely unlikely that a(4) exists. The term a(1)=1^3+1 is special. For other terms, note that b^k+1 can only be prime if k is a power of 2, so say k=2^p. Otherwise, if k has an odd factor, b^k+1 is algebraically factorizable. Therefore terms a(2) and later are of form (2^p-2)^(2^p)+1 with the additional obstruction that 2^p-1 must be a prime (namely a Mersenne prime), meaning that p (the exponent of the exponent) must be prime. With generalized Fermat numbers b^(2^p)+1 already primality tested to a high limits, the first undecided possibility for a(4) is 2147483646^2147483648+1 (has j=105097565 and p=31). - Jeppe Stig Nielsen, Sep 15 2022