1.Introduction

A.S. Janfada and  A. Abbaspour[1] showed that equation x^6+6z^3=y^6+6w^3 has infinite non-trivial primitive integer solutions.

We show that x^5 + az^3 = y^5 + aw^3 has infinitely many integer solutions.
     
     
2.Theorem
     
If a = 16(1+3n^2) then, equation x^5 + az^3 = y^5 + aw^3 has infinitely many integer solutions.
n is arbitrary.
 
Proof.

x^5+az^3=y^5+aw^3......................................................................(1)

Substitute x=-t+p, y=t+p, z=t+q, w=-t+q to equation (1), we obtain

6aq^2 = 2t^4+(-2a+20p^2)t^2+10p^4......................................................(2)
Multiply 6a to both side and let v=6aq, then

v^2 = 12at^4+(-12a^2+120ap^2)t^2+60ap^4................................................(3)

Let a = 16p^2/(1+3n^2) and p = 1+3n^2, then equation (3) becomes equation (4).

Let V=6aq and U=t,then

V^2 = (192+576n^2)t^4+(24192n^4+51840n^6-1152-1152n^2)t^2
    +  86400n^4+259200n^6+960+14400n^2+388800n^8+233280n^10............................(4)

Since quartic equation (4) has a rational solution Q(U,V)=(1+3n^2,96n(1+3n^2)^2),
this quartic equation (4) is birationally equivalent to an elliptic curve below.

Y^2+16(27n^4+6n^2-1)YX/n+(147456n+1769472n^3+7962624n^5+15925248n^7+11943936n^9)Y
= X^3+64(567n^8+540n^6+162n^4+12n^2-1)X^2/(n^2)+(-7077888n^2-106168320n^4-637009920n^6-1911029760n^8
-2866544640n^10-1719926784n^12)X-62412703137792n^18-163461841551360n^16-186247431585792n^14-120202243080192n^12
-47772686352384n^10-11814750388224n^8-1712282664960n^6-114152177664n^4+1358954496n^2+452984832

Transformation is given, 
U = (62705664n^12+101523456n^10+64696320n^8+1728n^6X+19906560n^6+1152n^4X+2764800n^4+3Yn^3+73728n^2+192n^2X+Yn-12288)/(Yn)
V = (1051066368n^11Y-14155776n^2X-98304Yn-1146617856n^13Y-4299816960n^15Y+265420800Yn^7+35389440Yn^5+1179648Yn^3+476228616192n^10X
    +56184274944n^8X+1868562432n^6X-247726080n^4X+53924575511052288n^26+177180748107743232n^24+261063421124935680n^22+875888640n^9Y
    +2175134072832n^12X+5985345208320n^14X+9973855420416n^16X+3761479876608n^20X+864n^8X^3+94058496n^14X^2+9318563315712n^18X
    +152285184n^12X^2+110592n^4X^2+4147200n^6X^2+97044480n^10X^2-3224862720n^17Y+576n^6X^3+29859840n^8X^2+96n^4X^3
    +128391846454886400n^18+2219118333788160n^12+786432X+12870886335971328n^14+49264427010097152n^16+226825595403632640n^20-18432X^2n^2
    -43486543872n^4-391378894848n^6+9784472371200n^8+228956653486080n^10)/(n^3Y^2)
X = (192nV+23040n^2+138240n^4+414720n^6+1152n^3V+622080n^8+1728n^5V+373248n^10+82944n^4U+331776n^6U+373248n^8U-1536U+1536)/(U^2-2U-6n^2U+1+6n^2+9n^4)
Y = (12288-73728n^2U^2+89579520n^8U+43130880n^6U+7741440n^4U-442368n^5V-36864n^3V+12288U^2-64696320n^8U^2+368640n^2U-2759049216n^12
    -2794881024n^14-1209323520n^16-5971968n^10U-101523456n^10U^2-1990656n^7V-3981312n^9V-2985984n^11V-268738560n^12U-268738560n^14U-62705664n^12U^2
    -19906560n^6U^2-2764800n^4U^2-9289728n^4-92897280n^6-294912n^2-487710720n^8-1504935936n^10-24576U)/(-nU^3+3nU^2+9n^3U^2-3nU-18n^3U-27n^5U+n+9n^3+27n^5+27n^7)
  
The point corresponding to point Q is P(X,Y)=(-64(567n^8+540n^6+162n^4+12n^2-1)/(n^2), 1024(3645n^12+2430n^10-729n^8-972n^6-261n^4-18n^2+1)/(n^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
2Q(U,V)=((57n^2-1)(1+3n^2)/((3n-1)(3n+1)), 288n(303n^4+18n^2-1)(1+3n^2)^2/((3n-1)^2(3n+1)^2))
q = v/(6a) = 48n(303n^4+18n^2-1)(1+3n^2)^2/((3n-1)^2(3n+1)^2(16+48n^2))
(x,y,z,w)=(-48(1+3n^2)n^2(3n-1)^2(3n+1)^2,
           2(33n^2-1)(1+3n^2)(3n-1)^2(3n+1)^2,
           (3n+1)^3(3n-1)^3(513n^4-66n^2+1+909n^5+54n^3-3n)(1+3n^2),
           (3n+1)^3(3n-1)^3(-513n^4+66n^2-1+909n^5+54n^3-3n)(1+3n^2))


Q.E.D.


3.Example

(n,a)=(2,26): (-3057600)^5 + 26*41757420250^3 = 4172350^5 + 26*24044042750^3



4.Reference

[1].A.S. Janfada and  A. Abbaspour, On Diophantine equations X^6+ 6Z^3 = Y^6± 6W^3,
    International Journal of Pure and Applied Mathematics, VOL:105, NO:4,2015