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.

2, 6, 90, 960, 1974, 2430, 2730, 2736, 6006, 6096, 6306, 7014, 11934, 14190, 18276, 18486, 21204, 24906, 24984, 25200, 27210, 35700, 38556, 39306, 40860, 44694, 45654, 47124, 49524, 51246, 53220, 56700, 58176, 63330, 63960, 72996, 76650, 80394, 85560

Offset: 1

Author

T. D. Noe, Aug 27 2003

Keywords

• nonn

Comments

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.

From Ryan Bresler and Russell Jarrett, May 03 2019: (Start)

All these k, except k=2, are multiples of 6.

Proof:

Suppose k == 1 (mod 3); then we have

k^2 == 1 (mod 3),

k^2 + 1 == 2 (mod 3), and

k^2 + 1 + k == 0 (mod 3),

so k^2 + 1 + k cannot be prime if k == 1 (mod 3).

Now suppose k == 2 (mod 3); then

k^2 == 1 (mod 3),

k^2 + 1 == 2 (mod 3), and

k^2 + 1 - k == 0 (mod 3),

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).

Now suppose k == 0 (mod 3); then

k^2 == 0 (mod 3) and

k^2 + 1 == 1 (mod 3),

so k^2 + 1 + k == 1 (mod 3) and k^2 + 1 - k == 1 (mod 3).

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).

Finally, if k == 1 (mod 2) for k > 2, then we have

k^2 == 1 (mod 2), and

k^2 + 1 == 0 (mod 2),

so k^2 + 1 cannot be prime if k == 1 (mod 2).

Now suppose k == 0 (mod 2); then

k^2 + 1 == 1 (mod 2),

so k^2 + 1 + k == 1 (mod 2) and k^2 + 1 - k == 1 (mod 2).

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.

(End)

Examples

			6 is a term of this sequence because 31, 37 and 43 are primes.
		

References

• Paulo Ribenboim, The New Book of Prime Number Records, Springer, 1996, p. 391.
• Paulo Ribenboim, The Little Book of Bigger Primes, Springer, Second Edition, 2000, pp. 256-259.

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• Eric W. Weisstein, MathWorld: Schinzel's Hypothesis
• Wikipedia, Schinzel's hypothesis H

Crossrefs

Cf. A014574 (first degree solutions: average of twin primes).

Cf. A231612 (similar, but with fourth-degree cyclotomic polynomials).

Cf. A231613 (similar, but with sixth-degree cyclotomic polynomials).

Cf. A231614 (similar, but with eighth-degree cyclotomic polynomials).

Cf. A233512 (similar, but increasing number of cyclotomic polynomials).

Programs

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

Mathematica

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}]
Join[{2}, Select[Range[6,80000,6], And@@PrimeQ[{#^2+1,#^2-#+1,#^2+#+1}]&]] (* Harvey P. Dale, Apr 07 2013 *)

Extensions

Definition and comment revised by N. J. A. Sloane, Sep 23 2019