A225768 - OEIS

%I #35 Nov 02 2024 06:49:42

%S 0,1,1,2,1,18,1,2,0,2,1,54,1,28,3,2,1,18,1,2,399,26,1,6,5,2,21,0,1,

%T 288,1,4,3,2,105,6,1,2,33,2,1,546,1,2,3,2,1,6,35,2,51,20,1,12,5,28,9,

%U 4,1,18,1,4,63,2,0,18,1,2,3,28,1,6,1,2,15,2,35,24,1,12,3,4,1,42,115,2,111,2,1,18,91,6,3,2,3,6,1,28,3,2

%N Least k > 0 such that k^6 + n is prime, or 0 if k^6 + n is never prime.

%C Motivated by the "particularly poor polynomial" n^6+1091 (composite for n=1,...,3905) mentioned on Weisstein's page about prime generating polynomials.

%C We have a(n) = 0 if n is a cube n = g^3 with g > 1 because then k^6 + g^3 = (k^2 + g)*(k^4 - k^2*g + g^2), which can be prime only when n = g = 1. - _T. D. Noe_, Nov 18 2013

%C By the theorem of Brillhart, Filaseta and Odlyzko (see link), if a(n) > n > 1 then x^6 + n must be irreducible.  If x^6 + n is irreducible, the Bunyakovsky conjecture implies a(n) is finite. - _Robert Israel_, Apr 25 2016

%H Robert Israel, <a href="/A225768/b225768.txt">Table of n, a(n) for n = 0..10000</a>

%H J. Brillhart, M. Filaseta, and A. Odlyzko, <a href="https://dx.doi.org/10.4153%2FCJM-1981-080-0">On an irreducibility theorem of A. Cohn</a>, Canadian Journal of Mathematics 33(1981), 1055-1059.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Prime-GeneratingPolynomial.html">Prime-Generating Polynomial</a>.

%p f:= proc(n) local exact, x,k,F,nf,F1,C;

%p     iroot(n,3,exact);

%p     if exact and n > 1 then return 0 fi;

%p     if irreduc(x^6+n) then

%p        for k from 1+(n mod 2) by 2 do if isprime(k^6+n) then return k fi od

%p     else

%p        F:= factors(x^6+n)[2]; #

%p        F1:= map(t -> t[1],F);

%p        nf:= nops(F);

%p        C:= map(t -> op(map(rhs@op,{isolve(t^2-1)})),F1);

%p        for k in sort(convert(select(type,C,positive),list)) do

%p          if isprime(k^6+n) then return k fi

%p        od:

%p        0

%p     fi

%p end proc:

%p map(f, [$0..100]); # _Robert Israel_, Apr 25 2016

%t {0, 1}~Join~Table[If[IrreduciblePolynomialQ[x^6 + n], SelectFirst[Range[1 + Mod[n, 2], 10^3, 2], PrimeQ[#^6 + n] &], 0], {n, 2, 120}] (* _Michael De Vlieger_, Apr 25 2016, Version 10 *)

%o (PARI) {(a,b=6)->#factor(x^b+a)~==1 & for(n=1, 9e9, ispseudoprime(n^b+a)&return(n)); a==1&return(1);print1("/*"a":", factor(x^b+a)"*/")} /* For illustrative purpose only. The polynomial x^6+a is factored to avoid an infinite loop when it is composite. But there could be x such that this is prime, when all factors but one are 1 (not for exponent b=6, but, e.g., x=4 for exponent b=4), see A225766. */

%Y See A085099, A225765, A225767, A225769, A225770 for the k^2, k^3, ..., k^8 analogs.

%K nonn

%O 0,4

%A _M. F. Hasler_, Jul 25 2013