%I #20 Feb 16 2025 08:32:51

%S 129,2188,2315,16385,16512,18571,78126,78253,80312,94509,279937,

%T 280064,282123,296320,358061,823544,823671,825730,839927,901668,

%U 1103479,2097153,2097280,2099339,2113536,2175277,2377088,2920695

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

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

%C 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

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

%H R. L. Ekl, <a href="http://dx.doi.org/10.1090/S0025-5718-96-00768-5">Equal Sums of Four Seventh Powers</a>, Math. Comput. 65, 1755-1756, 1996.

%H R. L. Ekl, <a href="http://dx.doi.org/10.1090/S0025-5718-98-00979-X">New Results in Equal Sums of Like Powers</a>, Math. Comput. 67, 1309-1315, 1998.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DiophantineEquation7thPowers.html">Diophantine Equation: 7th Powers</a>

%e 129 = 1^7+2^7.

%t 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 *)

%o (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) }

%Y Cf. A003369, A155468 (8th powers).

%K nonn,changed

%O 1,1

%A _Cino Hilliard_, Nov 22 2003

%E Edited by _Don Reble_, May 03 2006