A246119 a(n) is the least k such that k^(2^n)*(k^(2^n) - 1) + 1 is prime.

2, 2, 2, 5, 4, 2, 5, 196, 14, 129, 424, 484, 22, 5164, 7726, 13325, 96873, 192098, 712012, 123447

Offset: 0

Author

Serge Batalov, Aug 14 2014

Keywords

• nonn
• more
• hard

Comments

Numbers of the form k^m*(k^m-1)+1 with m > 0, k > 1 may be primes only if m is 3-smooth, because these numbers are Phi(6,k^m) and cyclotomic factorizations apply to any prime divisors >3. This series is a subset of A205506 with only m=2^n.

Trivially, a(n) <= a(n+1)^2. This upper bound, indeed, holds for a(4) = a(5)^2, a(7) = a(8)^2 and a(11) = a(12)^2.

The numbers of this form are Generalized Unique primes (see Links section).

a(16)=96873 corresponds to a prime with 653552 decimal digits.

The search for a(17) which corresponds to a 1385044-decimal digit prime was performed on a small Amazon EC2 cloud farm (40 GRID K520 GPUs), at a cost of approximately $1000 over three weeks.

a(18) <= 712012 corresponds to a prime with 3068389 decimal digits. - Serge Batalov, Jan 15 2018

a(19) <= 123447 corresponds to a prime with 5338805 decimal digits. - Serge Batalov, Jan 15 2018

a(20) <= 465859 corresponds to a prime with 11887192 decimal digits (not all lower candidates have been checked). This is the largest known non-Mersenne prime at the time of its discovery. - Serge Batalov, May 31 2023

Links

• C. Caldwell, Generalized unique primes
• C. Caldwell, The Prime Pages, Phi_3(-465859^1048576)

Crossrefs

Cf. A056993, A085398, A101406, A153436, A153438, A205506, A246120, A246121, A298206.

Programs

Mathematica

Table[SelectFirst[Range@ 200, PrimeQ[#^(2^n) (#^(2^n) - 1) + 1] &], {n, 0, 9}] (* Michael De Vlieger, Jan 15 2018 *)

PARI

a(n)=k=1;while(!ispseudoprime(k^(2^n)*(k^(2^n)-1)+1),k++);k
n=0;while(n<100,print1(a(n),", ");n++) \\ Derek Orr, Aug 14 2014

Formula

a(n) = A085398(3*2^(n+1)). - Jinyuan Wang, Jan 01 2023

Extensions

a(16) from Serge Batalov, Dec 30 2014

a(17) from Serge Batalov, Feb 10 2015

a(18-19) from Serge Batalov, May 31 2023