A088719 Numbers that can be represented as a^7 + b^7, with 0 < a < b, in exactly one way.

129, 2188, 2315, 16385, 16512, 18571, 78126, 78253, 80312, 94509, 279937, 280064, 282123, 296320, 358061, 823544, 823671, 825730, 839927, 901668, 1103479, 2097153, 2097280, 2099339, 2113536, 2175277, 2377088, 2920695

Offset: 1

Author

Cino Hilliard, Nov 22 2003

Keywords

• nonn

Comments

Conjecture: no number can be expressed as such a sum in more than one way.

No solutions to the 7.2.2 (A^7 + B^7 = C^7 + D^7), 7.2.3, 7.2.4, or 7.2.5 equations are known. The smallest 7.2.6 equation is: 125^7 + 24^7 = 121^7 + 94^7 + 83^7 + 61^7 + 57^7 + 27^7 = 476841744674549. - Jonathan Vos Post, May 04 2006

Examples

			129 = 1^7+2^7.
		

References

• Sastry, S. and Rai, T. "On Equal Sums of Like Powers." Math. Student 16, 18-19, 1948.

Links

• R. L. Ekl, Equal Sums of Four Seventh Powers, Math. Comput. 65, 1755-1756, 1996.
• R. L. Ekl, New Results in Equal Sums of Like Powers, Math. Comput. 67, 1309-1315, 1998.
• Eric Weisstein's World of Mathematics, Diophantine Equation: 7th Powers

Crossrefs

Cf. A003369, A155468 (8th powers).

Programs

Mathematica

lst={};e=7;Do[Do[x=a^e;Do[y=b^e;If[x+y==n,AppendTo[lst,n]],{b,Floor[(n-x)^(1/e)],a+1,-1}],{a,Floor[n^(1/e)],1,-1}],{n,3*8!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 23 2009 *)

PARI

powers2(m1,m2,p1) = { for(k=m1,m2, a=powers(k,p1); if(a==1,print1(k",")) ); } powers(n,p) = { z1=0; z2=0; c=0; cr = floor(n^(1/p)+1); for(x=1,cr, for(y=x+1,cr, z1=x^p+y^p; if(z1 == n,c++); ); ); return(c) }

Extensions

Edited by Don Reble, May 03 2006