A223934 Least prime p such that x^n-x-1 is irreducible modulo p.

2, 2, 2, 3, 2, 2, 7, 2, 17, 7, 5, 3, 3, 2, 109, 3, 101, 19, 229, 5, 2, 23, 23, 17, 107, 269, 2, 29, 2, 31, 37, 197, 107, 73, 37, 7, 59, 233, 3, 3, 7, 43, 43, 5, 2, 47, 269, 61, 43, 3, 53, 13, 3, 643, 13, 5, 151, 59, 2

Offset

2,1

Comments

Conjecture: a(n) < n*(n+3)/2 for all n>1.

Note that a(20) = 229 < 20*(20+3)/2 = 230.

The conjecture was motivated by E. S. Selmer's result that for any n>1 the polynomial x^n-x-1 is irreducible over the field of rational numbers.

We also conjecture that for every n=2,3,... there is a positive integer z not exceeding the (2n-2)-th prime such that z^n-z-1 is prime, and the Galois group of x^n-x-1 over the field of rationals is isomorphic to the symmetric group S_n.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 2..500

E. S. Selmer, On the irreducibility of certain trinomials, Math. Scand., 4 (1956) 287-302.

Example

a(8)=7 since f(x)=x^8-x-1 is irreducible modulo 7 but reducible modulo any of 2, 3, 5, for,
   f(x)==(x^2+x+1)*(x^6+x^5+x^3+x^2+1) (mod 2),
   f(x)==(x^3+x^2-x+1)*(x^5-x^4-x^3-x^2+x-1) (mod 3),
   f(x)==(x^2-2x-2)*(x^6+2x^5+x^4+x^3-x^2-2) (mod 5).

Mathematica

Do[Do[If[IrreduciblePolynomialQ[x^n-x-1, Modulus->Prime[k]]==True, Print[n, " ", Prime[k]]; Goto[aa]], {k, 1, PrimePi[n*(n+3)/2-1]}];
Print[n, " ", counterexample]; Label[aa]; Continue, {n, 2, 100}]

Crossrefs

Cf. A000040, A220072, A217785, A217788, A218465.

Cf. A002475 (n such that x^n-x-1 is irreducible over GF(2)).

Cf. A223938 (n such that x^n-x-1 is irreducible over GF(3)).

Sequence in context: A048288 A050677 A058013 * A237531 A238504 A031356

Adjacent sequences: A223931 A223932 A223933 * A223935 A223936 A223937

Keyword

nonn

Author

Zhi-Wei Sun, Mar 29 2013

Status

approved