A274035 Numbers n such that n^7 = a^2 + b^3 for positive integers a and b.

2, 5, 8, 9, 10, 12, 15, 17, 24, 26, 28, 31, 33, 36, 37, 40, 43, 44, 46, 50, 52, 54, 56, 57, 63, 65, 68, 69, 72, 73, 76, 80, 82, 89, 91, 98, 100, 101, 108, 113, 122, 126, 127, 128, 129, 134, 136, 141, 145, 148, 150, 152, 161, 164, 168, 170, 171, 174, 177, 183, 185, 189, 192, 196, 197

Offset

1,1

Links

Jean-François Alcover, Table of n, a(n) for n = 1..3001 (all terms from Charles R Greathouse IV except for a(58)=174)

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

Mathematica

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];
Reap[For[n = 1, n < 200, n++, If[okQ[n], Sow[n]]]][[2, 1]] (* Jean-François Alcover, Jan 30 2019 *)

Prog

(PARI) isA055394(n)=for(k=1, sqrtnint(n-1, 3), if(issquare(n-k^3), return(1))); 0
is(n)=isA055394(n^7)
(Sage) # Sage cannot handle n = 123, 174, ... without the fallback, even with descent_second_limit = 1000.
def fallback(n):
    return gp("my(n=" + str(n) + "); for(k=1, sqrtnint(n-1, 3), if(issquare(n-k^3), return(1))); 0")
def isA055394(z):
    z7 = z^7
    E = EllipticCurve([0, z7], descent_second_limit = 1000)
    try:
        for c in E.integral_points():
            if c[0] < 0 and c[1] != 0:
                return True
        return False
    except RuntimeError:
        return fallback(z7)
[x for x in range(1, 201) if isA055394(x)]

Crossrefs

Cf. A055394, A174115.

Sequence in context: A076871 A070049 A085254 * A068537 A047620 A283205

Adjacent sequences: A274032 A274033 A274034 * A274036 A274037 A274038

Keyword

nonn

Author

Charles R Greathouse IV, Jun 06 2016

Extensions

Missing term 174 inserted by Jean-François Alcover, Jan 30 2019

Status

approved