A376914 Values k for primitive solutions to k^5 + a^5 + b^5 + c^5 = d^5 + e^5 with k >= a >= b >= c > 0 and d >= e >= 0, repetitions allowed.
28, 37, 50, 63, 82, 86, 94, 99, 100, 102, 104, 112, 114, 129, 130, 133, 135, 137, 156, 172, 174, 184, 191, 196, 200, 213, 221, 236, 237, 241, 252, 258, 260, 270, 271, 279, 282, 291, 291, 291
Offset: 1
Examples
28^5 + 20^5 + 10^5 + 4^5 = 29^5 + 3^5 so 28 is a term. 133^5 + 110^5 + 84^5 + 27^5 = 144^5 + 0^5 so 133 is a term. 291^5 + 109^5 + 31^5 + 29^5 = 287^5 + 173^5 and 291^5 + 279^5 + 108^5 + 85^5 = 328^5 + 15^5 and 291^5 + 287^5 + 205^5 + 174^5 = 335^5 + 202^5 so 291 is included three times.
Links
- Andrew Howroyd, Solutions for k <= 300, Oct 2024.
- Eric Weisstein's World of Mathematics, Diophantine Equation-5th Powers.
- Wikipedia, Euler's sum of powers conjecture.
Programs
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Mathematica
aa = {}; Monitor[Do[Do[Do[Do[kk = PowersRepresentations[k^5 + a^5 + b^5 + c^5, 2, 5];If[kk != {}, If[GCD[k,a,b,c,kk[[1]][[1]],kk[[1]][[2]]]==1,Print[{k, a, b, c, kk}]; AppendTo[aa, k]]], {c, 1, b}], {b, 1, a}], {a, 1, k}], {k, 1, 200}], {a, k}]; aa -
PARI
lista(maxk, mink=1,prfull=0)={for(k=mink, maxk, for(a=1, k, for(b=1, a, for(c=1,b,my(s=k^5+a^5+b^5+c^5);for(d=sqrtnint((s-1)\2,5)+1, sqrtnint(s,5), my(e); if(ispower(s-d^5,5,&e) && gcd([k,a,b,c,d,e])==1, if(prfull, print([k,a,b,c,d,e]), print1(k, ", ") )) )))))} \\ Andrew Howroyd, Oct 09 2024
Extensions
a(26)-a(40) from Andrew Howroyd, Oct 09 2024
Comments