A088677 Numbers that can be represented as j^6 + k^6, with 0 < j < k, in exactly one way.

65, 730, 793, 4097, 4160, 4825, 15626, 15689, 16354, 19721, 46657, 46720, 47385, 50752, 62281, 117650, 117713, 118378, 121745, 133274, 164305, 262145, 262208, 262873, 266240, 277769, 308800, 379793, 531442, 531505, 532170, 535537, 547066

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.

Ekl (1996) has searched and found no solutions to the 6.2.2 Diophantine equation A^6 + B^6 = C^6 + D^6 with sums less than 7.25 * 10^26. - Jonathan Vos Post, May 04 2006

Examples

			65 = 1^6 + 2^6.
		

Links

• 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: 6th Powers.

Crossrefs

Cf. A003358, A088719 (7th powers).

Programs

Mathematica

lst={};e=6;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,2*8!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 22 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