A377496 Smallest prime p such that x^n - x - 1 splits modulo p.

5, 23, 83, 1973, 1151, 20959, 40609, 1627853, 57323489, 1616436271, 6814548563, 217642750067

Offset: 2

Author

Ben Whitmore, Oct 30 2024

Keywords

• nonn
• hard
• more

Comments

x^n - x - 1 is irreducible for all n (see link to Selmer, Theorem 1), and it appears that the Galois group is always the full symmetric group S_n.

For n > 3, it appears that all roots of x^n - x - 1 are distinct modulo a(n). For n = 2 and n = 3, there is a repeated root modulo a(n). The smallest primes modulo which x^2 - x - 1 and x^3 - x - 1 split with no repeated roots are 11 and 59 respectively.

Examples

			a(4) = 83 because x^4 - x - 1 has an irreducible factor of degree > 1 modulo all primes less than 83, but splits as (x + 3)(x + 7)(x + 14)(x + 59) modulo 83.
		

Links

• Ernst S. Selmer, On the irreducibility of certain trinomials, Mathematica Scandinavica 4 (1956), 287-302.

Crossrefs

Cf. A376950 (x^n + x + 1).

Programs

Mathematica

a[n_] := Module[{i},
 For[i = 1, True, i++,
  If[Total[Last /@ Rest[FactorList[x^n - x - 1, Modulus -> Prime[i]]]] == n,
   Return[Prime[i]];
  ]
 ]
];
a /@ Range[2, 8]