A279883 Primes of the form (prime(j)-1)^(prime(j)+1) + 1.

2, 17, 185302018885184100000000000000000000000000000001

Offset

1,1

Comments

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

Links

Table of n, a(n) for n=1..3.

Prog

(Magma) [(NthPrime(n)-1)^(NthPrime(n)+1)+1: n in[1..200] | IsPrime((NthPrime(n)-1)^(NthPrime(n)+1)+1)]
(PARI) print1(1^3+1, ", "); forprime(p=2, 19, if(isprime(2^p-1), a=(2^p-2)^(2^p)+1; ispseudoprime(a)&&print1(a, ", "))) \\ Jeppe Stig Nielsen, Sep 15 2022

Crossrefs

Cf. A000040, A279884.

Sequence in context: A112969 A208208 A290189 * A266166 A077452 A353309

Adjacent sequences: A279880 A279881 A279882 * A279884 A279885 A279886

Keyword

nonn,bref

Author

Jaroslav Krizek, Dec 21 2016

Status

approved