1.Introduction

Ajai Choudhry,etal.[1] showed that equation (x1^4 + x2^4 )(y1^4 + y2^4 ) = z1^4 + z2^4 has infinitely many parametric solutions.

We show that (x1^3 + x2^3 )(y1^3 + y2^3 ) = z1^3 + z2^3 has infinitely many parametric  solutions.
     
     
2.Theorem
     
Diophantine  equation (x1^3 + x2^3 )(y1^3 + y2^3 ) = z1^3 + z2^3 has infinitely many parametric  solutions.
    
 
Proof.

(x1^3 + x2^3 )(y1^3 + y2^3 ) = z1^3 + z2^3............................................(1)

Substitute x1=t+1, x2=t-1, y1=a, y2=1, z1=pt+a+1, z2=t-a-1 to equation (1), we obtain

(1-p^3+2a^3)t^2+(-3p^2a-3p^2+3a+3)t-6a-3p+6a^3+3-6pa-3pa^2-3a^2=0.....................(2)

To obtain the rational solution t of equation (2), the discriminant must be square as follows.

Let p=U, we obtain

V^2 = (-3a^2-3-6a)U^4
    +(-12a^2+24a^3-24a+12)U^3
    +(-18-36a-18a^2)Up^2
    +(48a^4+24a^3+24a+12a^2+24a^5+12)U
    -48a^6-48a^3+24a^5+48a^4+42a+21a^2-3..............................................(3)

Since quartic equation (3) has a rational solution Q(U,V)=( (2a^2-3a+1)/(a+1), 12a(a-2)(a^2-a+1)/(a+1) ),
this quartic equation (3) is birationally equivalent to an elliptic curve below.

Y^2+6(3a^3-8a^2+11a-2)YX/(a-2) = X^3-9(a^6+8a^5-22a^4+20a^3-7a^2+20a+4)X^2/((a-2)^2)+(1728a^8-10368a^7+25920a^6-38016a^5+36288a^4-20736a^3+6912a^2)X
                               -15552a^12-93312a^11+544320a^10-1337472a^9+1990656a^8-2270592a^7+1850688a^6-1337472a^5+544320a^4-186624a^3-62208a^2

Transformation is given, 

U = (-216a^9-1512a^8+6264a^7-10800a^6+10584a^5+24a^5X-10152a^4-120a^4X+2Ya^3+4968a^3+216a^3X-7Ya^2-192a^2X-3456a^2-864a+96aX+7Ya-2Y)/(Ya^2-Ya-2Y)

V = (1261578240a^14+99843840a^16-2603964672a^13-10450944a^17-559872a^18-442112256a^15+279936a^2X-158257152a^5+683790336a^6-102384Ya^11+31104aX+11943936a^3+7371648a^4X+8074080a^12X-22079952a^11X-1162512Ya^9+1879848Ya^8-2181168Ya^7+10368Ya+1953720Ya^6-1397736Ya^5+723168Ya^4-238464Ya^3-20736Ya^2+45089136a^10X-69186960a^9X+80598240a^8X-71436168a^7X+47753064a^6X-22672872a^5X+467208Ya^10+186624a^19-62856a^8X^2-2105352a^13X-18792a^15X-15552a^2X^2+118260a^7X^2-5184aX^2+340200a^14X+17172a^9X^2-972a^10X^2-95904a^4X^2+49248a^3X^2+131868a^5X^2-143532a^6X^2+3142748160a^8-1714701312a^7-768a^4X^3+12a^7X^3+396a^5X^3-324a^11X^2-108a^6X^3+864a^3X^3-576a^2X^3+192aX^3-4551199488a^9+5377197312a^10-5222486016a^11+4130922240a^12-938304a^3X+1944Ya^13+4536Ya^12)/(6a^2Y^2-5a^3Y^2+a^4Y^2+4aY^2-8Y^2)

X = (-144a^8+504a^7+216a^7U-576a^6U-1440a^6+24a^5V+792a^5U+2232a^5-1944a^4+72a^4U-48a^4V-576a^3U+1152a^3-216a^2+792a^2U+24a^2V-144aU-48aV+144a)/(1+4aU+2a^2U+4a^4-4a^3U-12a^3+13a^2+a^2U^2+U^2-6a-2U+2aU^2)

Y = (-267840a^5U+201744a^4U-1728aU+10368a^2U+4320a^13-108648a^4+297648a^5-586008a^6-78192a^3U+33912a^3-15552a^2-216a^6U^2-216a^8U^2+4608a^2V+5184a^2U^2-11520a^3V-6336a^5V-7488a^6V+97632a^7U-244080a^8U-576a^10V+216a^11U^2-6048a^12U+864a+2808a^3U^2+14976a^4V+136944a^6U+4752a^5U^2+3672a^4U^2+34128a^11U+209088a^9U-105840a^10U+4752a^7U^2-3024a^9U^2+1944a^10U^2-720792a^8+779760a^7+864aU^2+473904a^9-226152a^10+85752a^11-25920a^12+14400a^7V-10944a^8V+4032a^9V)/(-2+48a^5U-51a^4U-33aU+57a^2U-195a^4+138a^5-52a^6-15a^3U+2U^3+159a^3-75a^2+15a^2U^2-6U^2-a^4U^3+3a^2U^3+5aU^3-a^3U^3+19a-15a^3U^2-12a^6U+6U+6a^5U^2-9a^4U^2+8a^7+9aU^2)
    
The point corresponding to point Q is P(X,Y)=( 9(a^6+8a^5-22a^4+20a^3-7a^2+20a+4)/((a-2)^2), -54(3a^9+16a^8-119a^7+322a^6-439a^5+380a^4-265a^3+202a^2+4a-8)/((a-2)^3) ).

This point P is of infinite order, and the multiples mP, m = 2, 3, ...give infinitely many points.

Hence we can obtain infinitely many integer solutions for equation (1).

Case : m=2

x1 = 95a^15+446a^14-8631a^13+40778a^12-99202a^11+142956a^10-115374a^9+19812a^8+46971a^7+23366a^6-147067a^5+143442a^4-45448a^3+10736a^2+3600a-96
x2 = -363a^15+2946a^14-13077a^13+34734a^12-45366a^11-13260a^10+152310a^9-272340a^8+281961a^7-194214a^6+63039a^5+38502a^4-59928a^3+36048a^2+816a+480
y1 = a
y2 = 1
z1 = -(a+1)(3a^3-8a^2+11a-2)(109a^12-896a^11+2676a^10-5016a^9+10446a^8-23016a^7+33900a^6-29448a^5+14493a^4-3224a^3-776a^2-2016a-48)
z2 = -(a+1)(229a^15-1116a^14+393a^13+15706a^12-77358a^11+200832a^10-321414a^9+315180a^8-160335a^7-12836a^6+101997a^5-112566a^4+81304a^3-34032a^2-624a-480)

a is arbitrary.

Q.E.D.


3.Reference

[1]. Ajai Choudhry,Iliya Bluskov and Alexander James,  A quartic diophantine equation inspired by Brahmaguptafs identity
https://arxiv.org/abs/2010.08133





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