

A087277


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


6



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
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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 firstdegree cyclotomic polynomials, x1 and x+1, yield the twin primes for the numbers in A014574.
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+12 = 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. 256259.


LINKS



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[1x+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^2m+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 fourthdegree cyclotomic polynomials).
Cf. A231613 (similar, but with sixthdegree cyclotomic polynomials).
Cf. A231614 (similar, but with eighthdegree cyclotomic polynomials).
Cf. A233512 (similar, but increasing number of cyclotomic polynomials).


KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



