%I A057683
%S A057683 1,2,5,6,12,69,77,131,162,426,701,792,1221,1494,1644,1665,2129,2429,
%T A057683 2696,3459,3557,3771,4350,4367,5250,5670,6627,7059,7514,7929,8064,9177,
%U A057683 9689,10307,10431,11424,13296,13299,13545,14154,14286,14306,15137
%N A057683 Numbers n such that n^2+n+1, n^3+n+1 and n^4+n+1 are all prime.
%C A057683 After a(0) = 1, it is never the case that n^5 + n + 1 is prime. Proof: 
               consider integers modulo 4, that is, as 4n+k. (4*n+k)^5 + (4*n+k) 
               + 1 factors into irreducible components over Z. 1024n^5 + 1280k(n^4) 
               + 640(k^2)(n^3) + 160(k^3) (n^2) + (20(k^4)+4)n + (k^5+k+1) = (16n^2 
               + 8kn + 4n + k^2 + k + 1) (64n^3 + 48k(n^2) - 16n^2 + 12(k^2)n - 
               8kn + k^3 - k^2 + 1). - Jonathan Vos Post (jvospost3(AT)gmail.com), 
               Oct 17 2007
%e A057683 5 is included because 5^2+5+1=31, 5^3+5+1=131 and 5^4+5+1=631 are all 
               prime.
%Y A057683 Cf. A049407.
%Y A057683 Sequence in context: A108365 A064765 A082552 this_sequence A069480 A100613 
               A070911
%Y A057683 Adjacent sequences: A057680 A057681 A057682 this_sequence A057684 A057685 
               A057686
%K A057683 easy,nice,nonn
%O A057683 1,2
%A A057683 Harvey P. Dale (hpd1(AT)is2.nyu.edu), Oct 20 2000