

A246119


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


4



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

0,1


COMMENTS

Numbers of the form k^m*(k^m1)+1 with m > 0, k > 1 may be primes only if m is 3smooth, 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 1385044decimal 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 nonMersenne prime at the time of its discovery.  Serge Batalov, May 31 2023


LINKS



FORMULA



MATHEMATICA

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


PROG

(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


CROSSREFS



KEYWORD

nonn,more,hard


AUTHOR



EXTENSIONS



STATUS

approved



