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.

66, 67, 74, 83, 107, 118, 119, 123, 136, 142, 152, 155, 169, 170, 181, 182, 186, 201, 204, 215, 216, 224, 229, 233, 234, 248, 258, 264, 274, 282, 283, 286, 288, 289, 293, 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

Offset: 1

Author

Artur Jasinski, Sep 19 2024

Keywords

• nonn

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.

Examples

			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.
		

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.

Crossrefs

Cf. A063923, A345010, A376914.

Programs

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

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