A153442 Numbers k such that k^81*(k^81+1)+1 is prime.

1, 209, 210, 842, 1176, 1358, 1370, 1608, 1707, 1845, 1850, 2594, 2880, 2882, 3123, 3384, 4085, 4457, 4469, 4808, 5090, 5186, 5516, 5529, 5867, 5991, 6123, 6144, 6606, 6906, 7001, 7019, 7119, 7430, 7541, 7719, 8031, 8463, 8471, 8486, 8595, 8609, 8627

Offset: 1

Author

Pierre CAMI, Dec 26 2008

Keywords

• nonn

Comments

It seems numbers of the form k^n*(k^n+1)+1 with n > 0, k > 1 may be primes only if n has the form 3^j. When n is even, k^(4*n)+k^(2*n)+1=(k^(2*n)+1)^2-(k^n)^2=(k^(2*n)+k^n+1)*(k^(2*n)-k^n+1) so composite. But why if n odd > 3 and not a power of 3, k^n*(k^n+1)+1 is always composite?

Links

• Pierre CAMI, Table of n, a(n) for n=1,...,9439

Crossrefs

Cf. A153438.

Programs

Mathematica

k81Q[k_]:=Module[{k81=k^81},PrimeQ[k81(k81+1)+1]]; Select[Range[9000], k81Q] (* Harvey P. Dale, Aug 28 2011 *)
Select[Range[9000], PrimeQ[(#^81 (#^81 + 1)) + 1] &] (* Vincenzo Librandi, Jan 17 2015 *)