A377496 - OEIS

%I #11 Nov 03 2024 09:31:45

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

%T 217642750067

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

%C 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.

%C 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.

%H Ernst S. Selmer, <a href="https://doi.org/10.7146/math.scand.a-10478">On the irreducibility of certain trinomials</a>, Mathematica Scandinavica 4 (1956), 287-302.

%e 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.

%t a[n_] := Module[{i},

%t  For[i = 1, True, i++,

%t   If[Total[Last /@ Rest[FactorList[x^n - x - 1, Modulus -> Prime[i]]]] == n,

%t    Return[Prime[i]];

%t   ]

%t  ]

%t ];

%t a /@ Range[2, 8]

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

%K nonn,hard,more

%O 2,1

%A _Ben Whitmore_, Oct 30 2024