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
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Mar 29 2013
STATUS
approved