A231612 -id:A231612 - cp's OEIS frontend

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

T. D. Noe, Aug 27 2003

nonn

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)

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

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.

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

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).

[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

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 *)

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

A231613 Numbers n such that the four sixth-degree cyclotomic polynomials are simultaneously prime.

Original entry on oeis.org

32034, 162006, 339105, 458811, 1780425, 2989119, 2993100, 3080205, 4375404, 6129597, 6280221, 7565142, 8489820, 10268277, 11343741, 12065076, 13067295, 13333182, 15866508, 16472802, 17040537, 18028605, 19066758, 22633629, 24256362, 24365259, 25031349

Offset: 1

T. D. Noe, Dec 11 2013

nonn

The polynomials are cyclotomic(7,x) = 1 + x + x^2 + x^3 + x^4 + x^5 + x^6, cyclotomic(9,x) = 1 + x^3 + x^6, cyclotomic(14,x) = 1 - x + x^2 - x^3 + x^4 - x^5 + x^6, and cyclotomic(18,x) = 1 - x^3 + x^6. The numbers 7, 9, 14 and 18 are in the sixth row of A032447.

By Schinzel's hypothesis H, there are an infinite number of n 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.

See A087277.

Cf. A014574 (first degree solutions: average of twin primes).
Cf. A087277 (similar, but with second-degree cyclotomic polynomials).
Cf. A231612 (similar, but with fourth-degree cyclotomic polynomials).
Cf. A231614 (similar, but with eighth-degree cyclotomic polynomials).

t = {}; n = 0; While[Length[t] < 30, n++; If[PrimeQ[Cyclotomic[7, n]] && PrimeQ[Cyclotomic[9, n]] && PrimeQ[Cyclotomic[14, n]] && PrimeQ[Cyclotomic[18, n]], AppendTo[t, n]]]; t
Select[Range[251*10^5],AllTrue[Cyclotomic[{7,9,14,18},#],PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Oct 29 2016 *)

A231614 Numbers n such that the five eighth-degree cyclotomic polynomials are simultaneously prime.

Original entry on oeis.org

4069124, 8919014, 8942756, 46503870, 75151624, 82805744, 189326670, 197155324, 271490544, 365746304, 648120564, 1031944990

Offset: 1

T. D. Noe, Dec 11 2013

The polynomials are cyclotomic(15,x) = 1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8, cyclotomic(16,x) = 1 + x^8, cyclotomic(20,x) = 1 - x^2 + x^4 - x^6 + x^8, cyclotomic(24,x) = 1 - x^4 + x^8, and cyclotomic(30,x) = 1 + x - x^3 - x^4 - x^5 + x^7 + x^8. The numbers 15, 16, 20, 24 and 30 are in the eighth row of A032447.

See A087277.

Cf. A014574 (first degree solutions: average of twin primes).
Cf. A087277 (similar, but with second-degree cyclotomic polynomials).
Cf. A231612 (similar, but with fourth-degree cyclotomic polynomials).
Cf. A231613 (similar, but with sixth-degree cyclotomic polynomials).

t = {}; n = 0; While[Length[t] < 6, n++; If[PrimeQ[Cyclotomic[15, n]] && PrimeQ[Cyclotomic[16, n]] && PrimeQ[Cyclotomic[20, n]] && PrimeQ[Cyclotomic[24, n]] && PrimeQ[Cyclotomic[30, n]], AppendTo[t, n]]]; t

Extended to 12 terms by T. D. Noe, Dec 13 2013

A233512 The first n cyclotomic polynomials are simultaneously prime for these arguments.

Original entry on oeis.org

3, 4, 6, 150, 1068630, 6770610

Offset: 1

T. D. Noe, Dec 13 2013

The first six cyclotomic polynomials are x - 1, x + 1, x^2 + x + 1, x^2 + 1, x^4 + x^3 + x^2 + x + 1, and x^2 - x + 1.
By Schinzel's hypothesis H, this sequence is defined for all n.
a(7) > 2*10^9.

			At x = 3, x-1 = 2, which is prime. At x = 4, x-1 = 3 and x+1 = 5, which are both prime. At x = 6, x-1 = 5, x+1 = 7, and x^2+x+1 = 43, which are all prime.

See A087277.

Cf. A014574 (first degree solutions: average of twin primes).
Cf. A087277 (similar, but with second-degree cyclotomic polynomials).
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).

t = {}; n = 0; len = 0; While[len < 6, n++; found = True; i = 1; While[found && i <= len + 1, found = PrimeQ[Cyclotomic[i, n]]; i++]; If[found && i > len + 1, AppendTo[t, n]; len++]]; t

cp's OEIS Frontend

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.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

Extensions

A231613 Numbers n such that the four sixth-degree cyclotomic polynomials are simultaneously prime.

Views

Author

Keywords

Comments

References

Crossrefs

Programs

Mathematica

A231614 Numbers n such that the five eighth-degree cyclotomic polynomials are simultaneously prime.

Views

Author

Keywords

Comments

References

Crossrefs

Programs

Mathematica

Extensions

A233512 The first n cyclotomic polynomials are simultaneously prime for these arguments.

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Programs

Mathematica