A174115 Primitive solutions x of the Diophantine equation x^2 + y^3 = z^7, gcd(x,y,z) = 1.

-21063928, -15312283, -2213459, -71, -3, -1, 0, 1, 3, 71, 2213459, 15312283, 21063928

Offset: 1

Author

Michel Lagneau, Mar 08 2010

Keywords

• sign
• fini

Comments

The equation x^2 + y^3 = z^7 has several solutions, and some of them involve large integers. The set of primitive integer solutions is finite if we consider the general equation Ax^p + By^q = Cz^r and the relation 1/p + 1/q + 1/r - 1 = 1/2 + 1/3 + 1/7 - 1 < 0 (See Darmon and Grandville link).

Theorem: The primitive integer solutions to x^2 + y^3 = z^7 are the 16 triples: (-1,-1,0),(1,-1,0), (-1,0,1), (1,0,1), (0,1,1), (0,-1,-1) (-3,-2,1), (3,-2,1), (-71,-17,2), (71,-17,2), (-2213459,1414,65), (2213459,1414,65), (-15312283,9262,113), (15312283,9262,113), (-21063928,-76271,17), (21063928,-76271,17).

The proof is given in Darmon and Granville's paper. Remark: there exist other solutions where gcd(x,y,z) <> 1; for example: (8,4,2), (250,25,5), (729,162,9), (832,112,8), (3000,100,10), (3456,288,12), (12250,275,15), (14739,578,17), (19652,289,17).

Examples

			71^2 - 17^3 = 5041 - 4913 = 128 = 2^7;
2213459^2 + 1414^3 = 4899400744681 + 2827145944 = 4902227890625 = 65^7.
		

References

• L. J. Mordell, Diophantine Equations, Academic Press, NY, 1969, p. 205.

Links

• H. Darmon, A. Granville, On the equations z^m = F(x,y) and Ax^p + By^q = Cz^r , Bull. London Math. Soc. 27 (1995), 513-543.
• B. Poonen, E. Schaefer, M.Stoll, Twists of X (7) and primitive solutions to x^2+ y^3= z^7, Duke Math. J. 137 (2007), no. 1, 103-158.

Programs

Mathematica

(* This script is a recomputation of the x>0 solutions, assuming z max is 113 *) r[y_, z_]:= Reduce[x>0 && x^2+y^3==z^7, x, Integers]; Reap[Do[If[(rr=r[y, z]) =!= False, xx = rr[[2]]; If[GCD[xx, y, z] == 1, Print[{xx, y, z}]; Sow[{xx, y, z}]] ];  yy = -y; If[(rr=r[yy, z]) =!= False, xx = rr[[2]]; If[GCD[xx, yy, z]==1, Print[{xx, yy, z}];  Sow[{xx, yy, z}]]], {z, 0, 113}, {y, 0, 10^5}]][[2, 1]] (* Jean-François Alcover, Apr 10 2015 *)