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

1,1

Comments

Primitive means a solution has gcd(k,a,b,c,d,e) = 1.

In most of cases d > k.

This case is known in literature as 5.2.4 (see e.g. Eric Weisstein's World of Mathematics).

Links

Table of n, a(n) for n=1..40.

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.

Example

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.

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

Prog

(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

Crossrefs

Cf. A063923, A134341, A376289.

Sequence in context: A323005 A080983 A179105 * A096430 A204822 A179166

Adjacent sequences: A376911 A376912 A376913 * A376915 A376916 A376917

Keyword

nonn,more

Author

Artur Jasinski, Oct 09 2024

Extensions

a(26)-a(40) from Andrew Howroyd, Oct 09 2024

Status

approved