

A223934


Least prime p such that x^nx1 is irreducible modulo p.


10



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
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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^nx1 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 (2n2)th prime such that z^nz1 is prime, and the Galois group of x^nx1 over the field of rationals is isomorphic to the symmetric group S_n.


LINKS

ZhiWei Sun, Table of n, a(n) for n = 2..500
E. S. Selmer, On the irreducibility of certain trinomials, Math. Scand., 4 (1956) 287302.


EXAMPLE

a(8)=7 since f(x)=x^8x1 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^2x+1)*(x^5x^4x^3x^2+x1) (mod 3),
f(x)==(x^22x2)*(x^6+2x^5+x^4+x^3x^22) (mod 5).


MATHEMATICA

Do[Do[If[IrreduciblePolynomialQ[x^nx1, Modulus>Prime[k]]==True, Print[n, " ", Prime[k]]; Goto[aa]], {k, 1, PrimePi[n*(n+3)/21]}];
Print[n, " ", counterexample]; Label[aa]; Continue, {n, 2, 100}]


CROSSREFS

Cf. A000040, A220072, A217785, A217788, A218465.
Cf. A002475 (n such that x^nx1 is irreducible over GF(2)).
Cf. A223938 (n such that x^nx1 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

ZhiWei Sun, Mar 29 2013


STATUS

approved



