A274035 Numbers k such that k^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
Keywords
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.
Programs
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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 *) -
PARI
isA055394(n)=for(k=1,sqrtnint(n-1,3),if(issquare(n-k^3),return(1)));0 is(n)=isA055394(n^7)
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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)]
Extensions
Missing term 174 inserted by Jean-François Alcover, Jan 30 2019
Comments