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%I #17 Sep 08 2022 08:45:39
%S 1,2,11,44,45,56,62,63,110,170,219,234,245,261,263,333,395,398,402,
%T 413,428,434,437,498,557,558,578,633,692,695,723,731,750,761,774,794,
%U 797,804,806,846,854,855,863,906,923,926,977,1046,1085,1086
%N Numbers k such that k^9*(k^9+1)+1 is prime.
%C 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?
%H Pierre CAMI, <a href="/A153440/b153440.txt">Table of n, a(n) for n=1,...,38019</a>
%t k9pQ[n_]:=Module[{c=n^9},PrimeQ[c(c+1)+1]]; Select[Range[1200],k9pQ] (* _Harvey P. Dale_, Oct 29 2014 *)
%t Select[Range[1100], PrimeQ[(#^9 (#^9 + 1)) + 1] &] (* _Vincenzo Librandi_, Jan 17 2015 *)
%o (Magma) [n: n in [0..1100] | IsPrime(n^9*(n^9+1)+1)]; // _Vincenzo Librandi_, Jan 17 2015
%Y Cf. A153438.
%K nonn
%O 1,2
%A _Pierre CAMI_, Dec 26 2008
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