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

%I A377496 #11 Nov 03 2024 09:31:45
%S A377496 5,23,83,1973,1151,20959,40609,1627853,57323489,1616436271,6814548563,
%T A377496 217642750067
%N A377496 Smallest prime p such that x^n - x - 1 splits modulo p.
%C A377496 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 A377496 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 A377496 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 A377496 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 A377496 a[n_] := Module[{i},
%t A377496  For[i = 1, True, i++,
%t A377496   If[Total[Last /@ Rest[FactorList[x^n - x - 1, Modulus -> Prime[i]]]] == n,
%t A377496    Return[Prime[i]];
%t A377496   ]
%t A377496  ]
%t A377496 ];
%t A377496 a /@ Range[2, 8]
%Y A377496 Cf. A376950 (x^n + x + 1).
%K A377496 nonn,hard,more
%O A377496 2,1
%A A377496 _Ben Whitmore_, Oct 30 2024