A225767 Least k>0 such that k^5+n is prime, or 0 if k^5+n is never prime.

0, 1, 1, 8, 1, 2, 1, 4, 3, 2, 1, 2, 1, 6, 3, 2, 1, 6, 1, 10, 3, 2, 1, 14, 7, 4, 3, 2, 1, 2, 1, 22, 0, 8, 3, 2, 1, 4, 3, 2, 1, 2, 1, 10, 5, 4, 1, 2, 13, 10, 3, 2, 1, 6, 17, 12, 5, 2, 1, 12, 1, 12, 5, 4, 3, 2, 1, 4, 3, 2, 1, 2, 1, 4, 3, 2, 7, 2, 1, 4, 63, 2, 1, 18, 5, 4, 11, 32, 1, 14, 11, 6, 5, 4, 3, 2, 1, 6, 11, 2

Offset: 0

Author

M. F. Hasler, Jul 25 2013

Keywords

• nonn

Comments

See A225768 for motivation and references.

By the theorem of Brillhart, Filaseta and Odlyzko (see link), if a(n) > n > 1 then x^5 + n must be irreducible. If x^5 + n is irreducible, the Bunyakovsky conjecture implies a(n) is finite. - Robert Israel, Apr 25 2016

Examples

			a(3)=8 because 1^5+3, 2^5+3, ..., 7^5+3 are all composite, but 8^5+3=32771 is prime.
a(32)=0 because x^5+32 = (x + 2)(x^4 - 2x^3 + 4x^2 - 8x + 16) is composite for all integer values of x>0.
		

Links

• Robert Israel, Table of n, a(n) for n = 0..10000
• J. Brillhart, M. Filaseta, A. Odlyzko, On an irreducibility theorem of A. Cohn. Canad. J. Math. 33(1981), 1055-1059.

Crossrefs

See A085099, A225765--A225770 for the k^2, k^3, ..., k^8 analogs.

Programs

Maple

f:= proc(n) local x,k,F,nf,F1,C;
    if irreduc(x^5+n) then
       for k from 1+(n mod 2) by 2 do if isprime(k^5+n) then return k fi od
    else
       F:= factors(x^5+n)[2]; #
       F1:= map(t -> t[1],F);
       nf:= nops(F);
       C:= map(t -> op(map(rhs@op,{isolve(t^2-1)})),F1);
       for k in sort(convert(select(type,C,positive),list)) do
         if isprime(k^5+n) then return k fi
       od:
       0
    fi
end proc:
map(f, [$0..100]); # Robert Israel, Apr 25 2016

Mathematica

{0, 1}~Join~Table[If[IrreduciblePolynomialQ[x^5 + n], SelectFirst[Range[1 + Mod[n, 2], 10^2, 2], PrimeQ[#^5 + n] &], 0], {n, 2, 120}] (* Michael De Vlieger, Apr 25 2016, Version 10 *)

PARI

A225767(a,b=5)={#factor(x^b+a)~==1&for(n=1,9e9,ispseudoprime(n^b+a)&return(n));a==1 || print1("/*"factor(x^b+a)"*/")} \\ For illustrative purpose only. The polynomial is factored to avoid an infinite search loop when it is composite. But a factored polynomial can yield a prime when all factors but one equal 1. This happens for b=4, n=4, cf. A225766.