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A087277 Numbers k such that the three second-degree cyclotomic polynomials x^2 + 1, x^2 - x + 1 and x^2 + x + 1 are simultaneously prime when evaluated at x=k.

%I A087277 #56 Feb 16 2025 08:32:51
%S A087277 2,6,90,960,1974,2430,2730,2736,6006,6096,6306,7014,11934,14190,18276,
%T A087277 18486,21204,24906,24984,25200,27210,35700,38556,39306,40860,44694,
%U A087277 45654,47124,49524,51246,53220,56700,58176,63330,63960,72996,76650,80394,85560
%N A087277 Numbers k such that the three second-degree cyclotomic polynomials x^2 + 1, x^2 - x + 1 and x^2 + x + 1 are simultaneously prime when evaluated at x=k.
%C A087277 Schinzel's hypothesis H, if true, would imply that there are an infinite number of k that yield simultaneous primes. Note that the two first-degree cyclotomic polynomials, x-1 and x+1, yield the twin primes for the numbers in A014574.
%C A087277 From _Ryan Bresler_ and _Russell Jarrett_, May 03 2019: (Start)
%C A087277 All these k, except k=2, are multiples of 6.
%C A087277 Proof:
%C A087277 Suppose k == 1 (mod 3); then we have
%C A087277   k^2 == 1 (mod 3),
%C A087277   k^2 + 1 == 2 (mod 3), and
%C A087277   k^2 + 1 + k == 0 (mod 3),
%C A087277 so k^2 + 1 + k cannot be prime if k == 1 (mod 3).
%C A087277 Now suppose k == 2 (mod 3); then
%C A087277   k^2 == 1 (mod 3),
%C A087277   k^2 + 1 == 2 (mod 3), and
%C A087277   k^2 + 1 - k == 0 (mod 3),
%C A087277 so k^2 + 1 - k cannot be prime if k == 2 (mod 3) (with the exception of k=2, which yields k^2 + 1 - k = 2^2 + 1 - 2 = 4+1-2 = 3, which is prime).
%C A087277 Now suppose k == 0 (mod 3); then
%C A087277   k^2 == 0 (mod 3) and
%C A087277   k^2 + 1 == 1 (mod 3),
%C A087277 so k^2 + 1 + k == 1 (mod 3) and k^2 + 1 - k == 1 (mod 3).
%C A087277 Therefore k^2 + 1, k^2 + 1 + k and k^2 + 1 - k can all be prime only if k=2 or k == 0 (mod 3).
%C A087277 Finally, if k == 1 (mod 2) for k > 2, then we have
%C A087277   k^2 == 1 (mod 2), and
%C A087277   k^2 + 1 == 0 (mod 2),
%C A087277 so k^2 + 1 cannot be prime if k == 1 (mod 2).
%C A087277 Now suppose k == 0 (mod 2); then
%C A087277   k^2 + 1 == 1 (mod 2),
%C A087277   so k^2 + 1 + k == 1 (mod 2) and k^2 + 1 - k == 1 (mod 2).
%C A087277 Therefore, for k > 2, k == 0 (mod 2) and k == 0 (mod 3) must be satisfied for k^2 + 1, k^2 + 1 + k and k^2 + 1 - k to all be prime.
%C A087277 (End)
%D A087277 Paulo Ribenboim, The New Book of Prime Number Records, Springer, 1996, p. 391.
%D A087277 Paulo Ribenboim, The Little Book of Bigger Primes, Springer, Second Edition, 2000, pp. 256-259.
%H A087277 T. D. Noe, <a href="/A087277/b087277.txt">Table of n, a(n) for n = 1..1000</a>
%H A087277 Eric W. Weisstein, <a href="https://mathworld.wolfram.com/SchinzelsHypothesis.html">MathWorld: Schinzel's Hypothesis</a>
%H A087277 Wikipedia, <a href="http://en.wikipedia.org/wiki/Schinzel&#39;s_hypothesis_H">Schinzel's hypothesis H</a>
%e A087277 6 is a term of this sequence because 31, 37 and 43 are primes.
%t A087277 x=0; Table[x=x+2; While[ !(PrimeQ[1+x^2] && PrimeQ[1+x+x^2] && PrimeQ[1-x+x^2]), x=x+2]; x, {50}]
%t A087277 Join[{2}, Select[Range[6,80000,6], And@@PrimeQ[{#^2+1,#^2-#+1,#^2+#+1}]&]] (* _Harvey P. Dale_, Apr 07 2013 *)
%o A087277 (Magma) [m:m in [1..90000]| IsPrime(m^2+1) and IsPrime(m^2-m+1) and IsPrime(m^2+m+1) ]; // _Marius A. Burtea_, May 07 2019
%Y A087277 Cf. A014574 (first degree solutions: average of twin primes).
%Y A087277 Cf. A231612 (similar, but with fourth-degree cyclotomic polynomials).
%Y A087277 Cf. A231613 (similar, but with sixth-degree cyclotomic polynomials).
%Y A087277 Cf. A231614 (similar, but with eighth-degree cyclotomic polynomials).
%Y A087277 Cf. A233512 (similar, but increasing number of cyclotomic polynomials).
%K A087277 nonn
%O A087277 1,1
%A A087277 _T. D. Noe_, Aug 27 2003
%E A087277 Definition and comment revised by _N. J. A. Sloane_, Sep 23 2019