A274035 - OEIS

A274035

Numbers n such that n^7 = a^2 + b^3 for positive integers a and b.

%I #25 Feb 28 2020 02:03:29

%S 2,5,8,9,10,12,15,17,24,26,28,31,33,36,37,40,43,44,46,50,52,54,56,57,

%T 63,65,68,69,72,73,76,80,82,89,91,98,100,101,108,113,122,126,127,128,

%U 129,134,136,141,145,148,150,152,161,164,168,170,171,174,177,183,185,189,192,196,197

%N Numbers n such that n^7 = a^2 + b^3 for positive integers a and b.

%H Jean-François Alcover, <a href="/A274035/b274035.txt">Table of n, a(n) for n = 1..3001</a> (all terms from Charles R Greathouse IV except for a(58)=174)

%H Bjorn Poonen, Edward F. Schaefer, and Michael Stoll, <a href="http://arxiv.org/abs/math/0508174">Twists of X(7) and primitive solutions to x^2+y^3=z^7</a>, arXiv:math/0508174 [math.NT], 2005; Duke Math. J. 137:1 (2007), pp. 103-158.

%t okQ[n_] := Module[{a, b}, For[b = 1, b < n^(7/3), b++, If[IntegerQ[a = Sqrt[n^7 - b^3]] && a > 0, Print["n = ", n, ", a = ", a, ", b = ", b]; Return[True]]]; False];

%t Reap[For[n = 1, n < 200, n++, If[okQ[n], Sow[n]]]][[2, 1]] (* _Jean-François Alcover_, Jan 30 2019 *)

%o (PARI) isA055394(n)=for(k=1,sqrtnint(n-1,3),if(issquare(n-k^3),return(1)));0

%o is(n)=isA055394(n^7)

%o (Sage) # Sage cannot handle n = 123, 174, ... without the fallback, even with descent_second_limit = 1000.

%o def fallback(n):

%o return gp("my(n=" + str(n) + ");for(k=1,sqrtnint(n-1,3),if(issquare(n-k^3),return(1)));0")

%o def isA055394(z):

%o z7 = z^7

%o E = EllipticCurve([0,z7], descent_second_limit = 1000)

%o try:

%o for c in E.integral_points():

%o if c[0] < 0 and c[1] != 0:

%o return True

%o return False

%o except RuntimeError:

%o return fallback(z7)

%o [x for x in range(1, 201) if isA055394(x)]

%Y Cf. A055394, A174115.

%K nonn

%O 1,1

%A _Charles R Greathouse IV_, Jun 06 2016

%E Missing term 174 inserted by _Jean-François Alcover_, Jan 30 2019