OFFSET
1,1
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
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.
EXAMPLE
65 = 1^6 + 2^6.
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 *)
PROG
(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) }
CROSSREFS
KEYWORD
nonn
AUTHOR
Cino Hilliard, Nov 22 2003
EXTENSIONS
Edited by Don Reble, May 03 2006
STATUS
approved