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A225767 Least k>0 such that k^5+n is prime, or 0 if k^5+n is never prime. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

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

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.

EXAMPLE

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.

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 *)

PROG

(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.

CROSSREFS

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

Sequence in context: A271547 A010154 A109011 * A019763 A307506 A179050

Adjacent sequences:  A225764 A225765 A225766 * A225768 A225769 A225770

KEYWORD

nonn

AUTHOR

M. F. Hasler, Jul 25 2013

STATUS

approved

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Last modified January 25 03:49 EST 2020. Contains 331241 sequences. (Running on oeis4.)