OFFSET
1,1
COMMENTS
This case is known in literature as 5.3.3 (see e.g. Eric Weisstein's World of Mathematics).
Primitive means a solution has gcd(k,a,b,c,d,e) = 1.
For primitive solutions of the 5.1.5 case see A063923.
For primitive solutions of the 5.2.4 case see A376914.
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.
In every known case, k+a+b-c-d-e is even and very often zero.
This sequence is infinite as follows:
1) Bremner's modified one parameter identity (with conditions k+a+b-c-d-e=0 and k-a=c-d):
(37888 + 67978*w + 53683*w^2 + 24217*w^3 + 6750*w^4 + 1164*w^5 + 115*w^6 + 5*w^7)^5+
(15744 + 33046*w + 29861*w^2 + 15193*w^3 + 4738*w^4 + 912*w^5 + 101*w^6 + 5*w^7)^5+
(16376 + 33534*w + 29739*w^2 + 14937*w^3 + 4622*w^4 + 888*w^5 + 99*w^6 + 5*w^7)^5
=
(27912 + 52390*w + 43165*w^2 + 20281*w^3 + 5882*w^4 + 1056*w^5 + 109*w^6 + 5*w^7)^5+
(5768 + 17458*w + 19343*w^2 + 11257*w^3 + 3870*w^4 + 804*w^5 + 95*w^6 + 5*w^7)^5+
(36328 + 64710*w + 50775*w^2 + 22809*w^3 + 6358*w^4 + 1104*w^5 + 111*w^6 + 5*w^7)^5
which generate members of this sequence for nonnegative w=0,1,2,3,...
2) Moessner's one parameter identity (k+a+b-c-d-e=40*n)
(a^36 + 8*a^26 + 12*a^16 + 20*a^11 - a^6)^5+
(a^33 - 12*a^23 - 28*a^13 - a^3)^5+
(a^30 + 20*a^25 - 12*a^20 - 8*a^10 - 1)^5
=
(a^36 + 8*a^26 + 12*a^16 - 20*a^11 - a^6)^5+
(a^33 + 28*a^23 + 12*a^13 - a^3)^5+
(a^30 - 20*a^25 - 12*a^20 - 8*a^10 - 1)^5
which generate members of this sequence for a=2,3,4,...
3) Moessner's two parameter identity (with condition k+a+b-c-d-e=0):
(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+
(-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+
(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
=
(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+
(-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+
(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
4) Choudhry and Wróblewski two parameter identity:
(2 p^15 q + 6 p^5 q^11)^5 +
(p^16 - 3 p^11 q^5 - 5 p^6 q^10 - p q^15)^5 +
(6 p^11 q^5 + 2 p q^15)^5
= (p^16 + 3 p^11 q^5 - 5 p^6 q^10 + p q^15)^5 +
(p^15 q + 5 p^10 q^6 + 3 p^5 q^11 - q^16)^5 +
(p^15 q - 5 p^10 q^6 + 3 p^5 q^11 + q^16)^5
5) Edward Brisse two parameter identity (with condition k+a+b-c-d-e=0):
(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+
(-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+
(a^9-22*a^5*b^4-125*a^3*b^6-79*a*b^8)^5
=
(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+
(-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+
(a^9-22*a^5*b^4+175*a^3*b^6+521*a*b^8)^5
When we take b=1 in this identity we obtain the Lander 1968 one parameter identity.
LINKS
A. Bremner, A geometric approach to equal sums of fifth powers, Number Th. 13, 337-354, 1981.
Edward Brisse in Jean-Charles Meyrignac Identities Of Equal Sums Of Like Power, Computing Minimal Equal Sums Of Like Powers 2001.
A. Choudhry and J. Wróblewski, A quintic diophantine equation with applications to two diophantine systems concerning fifth powers, Rocky Mountain J. Math. 43(6): 1893-1899 (2013).
Andrew Howroyd, Solutions for k <= 500, Oct 2024.
L. J. Lander, Geometric aspects of diophantine equations involving equal sums of like powers, American Mathematical Monthly Volume 75 no 6-10 1968 pp. 1061-1073
L. J. Lander, T. R. Parkin, and J. L. Selfridge, A Survey of Equal Sums of Like Powers, Math. Comput. 21, 446-459, 1967 (Table VI).
A. Moessner, Due sistemi diofantei., Bollettino dell'Unione Matematica Italiana Serie 3 6 (1951), p. 117-118 (Italian) Sezione Scientifica
Eric Weisstein's World of Mathematics, Diophantine Equation-5th Powers.
Wikipedia, Euler's sum of powers conjecture.
EXAMPLE
67^5 + 28^5 + 24^5 = 62^5 + 54^5 + 3^5 so 67 is a term.
399^5 + 237^5 + 62^5 = 382^5 + 307^5 + 9^5 so 399 is a term.
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.
MATHEMATICA
ww = {}; Monitor[Do[Do[Do[Do[kk = PowersRepresentations[e^5 + d^5 + c^5 - k^5, 2, 5];
If[kk != {}, If[GCD[k, c, d, e, kk[[1]][[1]], kk[[1]][[2]]] == 1,
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
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Artur Jasinski, Sep 19 2024
STATUS
approved