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%I #10 Sep 15 2022 11:46:12
%S 2,17,185302018885184100000000000000000000000000000001
%N Primes of the form (prime(j)-1)^(prime(j)+1) + 1.
%C Prime terms from A279884.
%C If a(4) exists, it must be bigger than (prime(2200)-1)^(prime(2200)+1) + 1 = 19422^19424 + 1.
%C Corresponding pairs of numbers (j, prime(j)): (1, 2); (2, 3); (11, 31); ...
%C 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
%o (Magma) [(NthPrime(n)-1)^(NthPrime(n)+1)+1: n in[1..200] | IsPrime((NthPrime(n)-1)^(NthPrime(n)+1)+1)]
%o (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
%Y Cf. A000040, A279884.
%K nonn,bref
%O 1,1
%A _Jaroslav Krizek_, Dec 21 2016
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