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A376289 Values k for primitive solutions to k^5 + a^5 + b^5 = c^5 + d^5 + e^5 with k >= a >= b >= 0 and k > c >= d >= e >= 0, repetitions allowed.

%I A376289 #111 Oct 29 2024 23:32:58
%S A376289 66,67,74,83,107,118,119,123,136,142,152,155,169,170,181,182,186,201,
%T A376289 204,215,216,224,229,233,234,248,258,264,274,282,283,286,288,289,293,
%U A376289 294,307,310,310,328,331,348,364,364,373,377,378,394,399,413,417,420,421,425,426,430,430,433,436,448,459,470,474,480,486,490,498
%N A376289 Values k for primitive solutions to k^5 + a^5 + b^5 = c^5 + d^5 + e^5 with k >= a >= b >= 0 and k > c >= d >= e >= 0, repetitions allowed.
%C A376289 This case is known in literature as 5.3.3 (see e.g. Eric Weisstein's World of Mathematics).
%C A376289 Primitive means a solution has gcd(k,a,b,c,d,e) = 1.
%C A376289 For primitive solutions of the 5.1.5 case see A063923.
%C A376289 For primitive solutions of the 5.2.4 case see A376914.
%C A376289 Although the definition does not require all coefficients to be nonzero or distinct, all known solutions have k > a > b > 0 and c > d > e > 0.
%C A376289 In every known case, k+a+b-c-d-e is even and very often zero.
%C A376289 This sequence is infinite as follows:
%C A376289 1) Bremner's modified one parameter identity (with conditions k+a+b-c-d-e=0 and k-a=c-d):
%C A376289   (37888 + 67978*w + 53683*w^2 + 24217*w^3 + 6750*w^4 + 1164*w^5 + 115*w^6 + 5*w^7)^5+
%C A376289   (15744 + 33046*w + 29861*w^2 + 15193*w^3 + 4738*w^4 + 912*w^5 + 101*w^6 + 5*w^7)^5+
%C A376289   (16376 + 33534*w + 29739*w^2 + 14937*w^3 + 4622*w^4 + 888*w^5 + 99*w^6 + 5*w^7)^5
%C A376289   =
%C A376289   (27912 + 52390*w + 43165*w^2 + 20281*w^3 + 5882*w^4 + 1056*w^5 + 109*w^6 + 5*w^7)^5+
%C A376289   (5768 + 17458*w + 19343*w^2 + 11257*w^3 + 3870*w^4 + 804*w^5 + 95*w^6 + 5*w^7)^5+
%C A376289   (36328 + 64710*w + 50775*w^2 + 22809*w^3 + 6358*w^4 + 1104*w^5 + 111*w^6 + 5*w^7)^5
%C A376289   which generate members of this sequence for nonnegative w=0,1,2,3,...
%C A376289 2) Moessner's one parameter identity (k+a+b-c-d-e=40*n)
%C A376289   (a^36 + 8*a^26 + 12*a^16 + 20*a^11 - a^6)^5+
%C A376289   (a^33 - 12*a^23 - 28*a^13 - a^3)^5+
%C A376289   (a^30 + 20*a^25 - 12*a^20 - 8*a^10 - 1)^5
%C A376289   =
%C A376289   (a^36 + 8*a^26 + 12*a^16 - 20*a^11 - a^6)^5+
%C A376289   (a^33 + 28*a^23 + 12*a^13 - a^3)^5+
%C A376289   (a^30 - 20*a^25 - 12*a^20 - 8*a^10 - 1)^5
%C A376289   which generate members of this sequence for a=2,3,4,...
%C A376289 3) Moessner's two parameter identity (with condition k+a+b-c-d-e=0):
%C A376289   (75*x^7-230*x^6*y-113*x^5*y^2+510*x^4*y^3-407*x^3*y^4+62*x^2*y^5+125*x*y^6-150*y^7)^5+
%C A376289   (-175*x^7+170*x^6*y-391*x^5*y^2-30*x^4*y^3+451*x^3*y^4-602*x^2*y^5+115*x*y^6-50*y^7)^5+
%C A376289   (175*x^7-160*x^6*y+387*x^5*y^2-108*x^4*y^3+5*x^3*y^4-336*x^2*y^5+265*x*y^6-100*y^7)^6
%C A376289   =
%C A376289   (25*x^7-290*x^6*y+689*x^5*y^2-138*x^4*y^3+27*x^3*y^4-62*x^2*y^5+155*x*y^6-150*y^7)^5+
%C A376289   (-25*x^7-653*x^5*y^2+564*x^4*y^3-195*x^3*y^4-208*x^2*y^5+105*x*y^6-100*y^7)^5+
%C A376289   (75*x^7+70*x^6*y-153*x^5*y^2-54*x^4*y^3+217*x^3*y^4-606*x^2*y^5+245*x*y^6-50*y^7)^5
%C A376289 4) Choudhry and Wróblewski two parameter identity:
%C A376289   (2 p^15 q + 6 p^5 q^11)^5 +
%C A376289   (p^16 - 3 p^11 q^5 - 5 p^6 q^10 - p q^15)^5 +
%C A376289   (6 p^11 q^5 + 2 p q^15)^5
%C A376289   = (p^16 + 3 p^11 q^5 - 5 p^6 q^10 + p q^15)^5 +
%C A376289   (p^15 q + 5 p^10 q^6 + 3 p^5 q^11 - q^16)^5 +
%C A376289   (p^15 q - 5 p^10 q^6 + 3 p^5 q^11 + q^16)^5
%C A376289 5) Edward Brisse two parameter identity (with condition k+a+b-c-d-e=0):
%C A376289   (2*a^8*b+10*a^7*b^2-20*a^6*b^3+20*a^5*b^4-34*a^4*b^5-10*a^3*b^6+270*a^2*b^7-20*a*b^8+682*b^9)^5+
%C A376289   (-2*a^8*b+10*a^7*b^2+20*a^6*b^3+20*a^5*b^4+34*a^4*b^5-10*a^3*b^6-270*a^2*b^7-20*a*b^8-682*b^9)^5+
%C A376289   (a^9-22*a^5*b^4-125*a^3*b^6-79*a*b^8)^5
%C A376289    =
%C A376289   (a^8*b+10*a^7*b^2-10*a^6*b^3+20*a^5*b^4-92*a^4*b^5-160*a^3*b^6-15*a^2*b^7-320*a*b^8+341*b^9)^5+
%C A376289   (-a^8*b+10*a^7*b^2+10*a^6*b^3+20*a^5*b^4+92*a^4*b^5-160*a^3*b^6+15*a^2*b^7-320*a*b^8-341*b^9)^5+
%C A376289   (a^9-22*a^5*b^4+175*a^3*b^6+521*a*b^8)^5
%C A376289 When we take b=1 in this identity we obtain the Lander 1968 one parameter identity.
%H A376289 A. Bremner, <a href="https://doi.org/10.1016/0022-314X(81)90019-6">A geometric approach to equal sums of fifth powers</a>, Number Th. 13, 337-354, 1981.
%H A376289 Edward Brisse in Jean-Charles Meyrignac <a href="http://euler.free.fr/identities.htm#ref05">Identities Of Equal Sums Of Like Power</a>, Computing Minimal Equal Sums Of Like Powers 2001.
%H A376289 A. Choudhry and J. Wróblewski, <a href="https://projecteuclid.org/journals/rocky-mountain-journal-of-mathematics/volume-43/issue-6/A-quintic-diophantine-equation-with-applications-to-two-diophantine-systems/10.1216/RMJ-2013-43-6-1893.full">A quintic diophantine equation with applications to two diophantine systems concerning fifth powers</a>, Rocky Mountain J. Math. 43(6): 1893-1899 (2013).
%H A376289 Andrew Howroyd, <a href="/A376289/a376289.txt">Solutions for k <= 500</a>, Oct 2024.
%H A376289 L. J. Lander, <a href="https://doi.org/10.1080/00029890.1968.11971124">Geometric aspects of diophantine equations involving equal sums of like powers</a>, American Mathematical Monthly Volume 75 no 6-10 1968 pp. 1061-1073
%H A376289 L. J. Lander, T. R. Parkin, and J. L. Selfridge, <a href="https://doi.org/10.1090/S0025-5718-1967-0222008-0">A Survey of Equal Sums of Like Powers</a>, Math. Comput. 21, 446-459, 1967 (Table VI).
%H A376289 A. Moessner, <a href="https://www.bdim.eu/item?fmt=pdf&amp;id=BUMI_1951_3_6_2_117_0">Due sistemi diofantei.</a>, Bollettino dell'Unione Matematica Italiana Serie 3 6 (1951), p. 117-118 (Italian) Sezione Scientifica
%H A376289 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DiophantineEquation5thPowers.html">Diophantine Equation-5th Powers</a>.
%H A376289 Wikipedia, <a href="https://en.wikipedia.org/wiki/Euler%27s_sum_of_powers_conjecture">Euler's sum of powers conjecture</a>.
%e A376289 67^5 + 28^5 + 24^5 = 62^5 + 54^5 + 3^5 so 67 is a term.
%e A376289 399^5 + 237^5 + 62^5 = 382^5 + 307^5 + 9^5 so 399 is a term.
%e A376289 310^5 + 118^5 + 102^5 = 271^5 + 270^5 + 49^5 and 310^5 + 124^5 + 116^5 = 294^5 + 235^5 + 21^5 so 310 is a repeated term.
%t A376289 ww = {}; Monitor[Do[Do[Do[Do[kk = PowersRepresentations[e^5 + d^5 + c^5 - k^5, 2, 5];
%t A376289 If[kk != {},If[GCD[k, c, d, e, kk[[1]][[1]], kk[[1]][[2]]] == 1,
%t A376289 AppendTo[ww, k]; Print[k];Print[{k, kk[[1]][[2]], kk[[1]][[1]], c, d, e}]]], {e, 0, d}],{d, 0, c}], {c, 0, k - 1}], {k, 4, 186}], {c, k}];ww
%o A376289 (PARI) lista(maxk, prfull=0)={for(k=1, maxk, for(a=0, k, for(b=0, a, my(s=k^5+a^5+b^5); for(c=sqrtnint(s\3,5), k-1, for(d=sqrtnint((s-c^5-1)\2,5)+1, min(c, sqrtnint(s-c^5,5)), my(e); if(ispower(s-c^5-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 08 2024
%Y A376289 Cf. A063923, A345010, A376914.
%K A376289 nonn
%O A376289 1,1
%A A376289 _Artur Jasinski_, Sep 19 2024