OFFSET
1,1
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)
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
EXAMPLE
6 is a term of this sequence because 31, 37 and 43 are primes.
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 *)
PROG
(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
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).
KEYWORD
nonn
AUTHOR
T. D. Noe, Aug 27 2003
EXTENSIONS
Definition and comment revised by N. J. A. Sloane, Sep 23 2019
STATUS
approved