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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.

%I A376914 #15 Oct 29 2024 23:34:04
%S A376914 28,37,50,63,82,86,94,99,100,102,104,112,114,129,130,133,135,137,156,
%T A376914 172,174,184,191,196,200,213,221,236,237,241,252,258,260,270,271,279,
%U A376914 282,291,291,291
%N 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.
%C A376914 Primitive means a solution has gcd(k,a,b,c,d,e) = 1.
%C A376914 In most of cases d > k.
%C A376914 This case is known in literature as 5.2.4 (see e.g. Eric Weisstein's World of Mathematics).
%H A376914 Andrew Howroyd, <a href="/A376914/a376914.txt">Solutions for k <= 300</a>, Oct 2024.
%H A376914 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DiophantineEquation5thPowers.html">Diophantine Equation-5th Powers</a>.
%H A376914 Wikipedia, <a href="https://en.wikipedia.org/wiki/Euler%27s_sum_of_powers_conjecture">Euler's sum of powers conjecture</a>.
%e A376914 28^5 + 20^5 + 10^5 + 4^5 = 29^5 + 3^5 so 28 is a term.
%e A376914 133^5 + 110^5 + 84^5 + 27^5 = 144^5 + 0^5 so 133 is a term.
%e A376914 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.
%t A376914 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
%o A376914 (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
%Y A376914 Cf. A063923, A134341, A376289.
%K A376914 nonn,more
%O A376914 1,1
%A A376914 _Artur Jasinski_, Oct 09 2024
%E A376914 a(26)-a(40) from _Andrew Howroyd_, Oct 09 2024