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

3, 4, 6, 150, 1068630, 6770610

Offset: 1

Author

T. D. Noe, Dec 13 2013

Keywords

• nonn
• hard
• more

Comments

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.

Examples

			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.
		

References

• See A087277.

Crossrefs

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).

Programs

Mathematica

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