A153440 Numbers k such that k^9*(k^9+1)+1 is prime.

1, 2, 11, 44, 45, 56, 62, 63, 110, 170, 219, 234, 245, 261, 263, 333, 395, 398, 402, 413, 428, 434, 437, 498, 557, 558, 578, 633, 692, 695, 723, 731, 750, 761, 774, 794, 797, 804, 806, 846, 854, 855, 863, 906, 923, 926, 977, 1046, 1085, 1086

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,...,38019

Crossrefs

Cf. A153438.

Programs

Magma

[n: n in [0..1100] | IsPrime(n^9*(n^9+1)+1)]; // Vincenzo Librandi, Jan 17 2015

Mathematica

k9pQ[n_]:=Module[{c=n^9},PrimeQ[c(c+1)+1]]; Select[Range[1200],k9pQ] (* Harvey P. Dale, Oct 29 2014 *)
Select[Range[1100], PrimeQ[(#^9 (#^9 + 1)) + 1] &] (* Vincenzo Librandi, Jan 17 2015 *)