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^6 + az^3 = y^6 + aw^3 has infinitely many integer solutions.
     
     
2.Theorem
     
If a=4256p^3/(25+3n^2) then, equation x^6 + az^3 = y^6 + aw^3 has infinitely many integer solutions.
n,p are arbitrary.
 
Proof.

x^6+az^3=y^6+aw^3........................................................................(1)

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

6aq^2 = 12pt^4-2t^2a+40t^2p^3+12p^5......................................................(2)
Multiply 6a to both side and let v=6aq, then

v^2 = 72apt^4+(-12a^2+240ap^3)t^2+72ap^5.................................................(3)

Let a = 4256p^3/(25+3n^2), then equation (3) becomes equation (4).

v^2 = 306432p^4(25t^4+3t^4n^2-626t^2p^2+10t^2p^2n^2+25p^4+3p^4n^2)/((25+3n^2)^2).........(4)

V^2 = (7660800+919296n^2)U^4+(-191826432p^2+3064320p^2n^2)U^2+7660800p^4+919296p^4n^2....(5)

Since quartic equation (5) has a rational solution Q(U,V)=(5p,25536p^2n),
this quartic equation (5) is birationally equivalent to an elliptic curve below.

Y^2+1920p(39+10n^2)YX/n+(7825047552000p^3n+939005706240p^3n^3)Y
= X^3+9216p^2*(25873n^2+5295n^4-152100)X^2/(n^2)+(-19982041428787200p^4n^2-2397844971454464n^4p^4)X
-1403441397308503307059200p^6n^2
-1546853550906189966999552p^6n^4-117011765365414381486080n^6p^6+28009898508151601233920000p^6.

Transformation is given, 
U = (51072p^2n^2X+12177892048896p^4n^2+2492248227840n^4p^4-71590359859200p^4+5pYn)/(Yn)
V = (2680343073128448000p^5Yn+52866727280640p^5n^7Y+182849892631921557504p^6n^8X-10397123031859200p^5n^5Y
  +  231327176337653760p^5Yn^3-34140808039902688051200p^6Xn^2+1698792498494114365440p^6n^6X-4083271464294989955072p^6n^4X
  -  107385539788800p^4X^2n^2+3738372341760p^4n^6X^2+18266838073344p^4n^4X^2+25536p^2n^4X^3+100352044657929093120000p^6X
  -  715260768304159289109381120000p^8n^4+2988012440371221645628538880p^8n^10+39500452275940466997300559872p^8n^8+35838279521669940449063731200p^8n^6)/(Y^2n^3)
X = (51072p^2nV-1147281408p^4n^2+1912135680p^3U-9560678400p^4+490291200p^3n^2U)/(U^2-10pU+25p^2)
Y = (-2608349184p^4n^3V-3712249626624p^6n^4+24122327040000p^5n^2U+183835666022400p^6n^2-117669888000p^5n^4U
  -  12177892048896p^4n^2U^2-2492248227840p^4n^4U^2+71590359859200p^4U^2-715903598592000p^5U+1789758996480000p^6)/(-nU^3+15npU^2-75nUp^2+125p^3n)

The point corresponding to point Q is P(X,Y)=(-9216p^2(25873n^2+5295n^4-152100)/n^2, -690094080p^3(13127n^2-590n^4+152100+3n^6)/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)=(-5(18119n^2+152100)p/(64873n^2-152100), 25536p^4n(531274129n^4-3993233400n^2-69403230000)/((64873n^2-152100)^2(25+3n^2)))
q = v/(6a) =pn(531274129n^4-3993233400n^2-69403230000)/((64873n^2-152100)^2)
(x,y,z,w)=(4p(38867n^2+152100), -18p(1429n^2+50700),
           p(-5877169435n^4-35556417000n^2+115672050000+531274129n^5-3993233400n^3-69403230000n),
           p(5877169435n^4+35556417000n^2-115672050000+531274129n^5-3993233400n^3-69403230000n))

Case : m=3
(x,y,z,w)=(4p(5590508909077n^6-236011797578700n^4-2697495340410000n^2-3518743761000000),
           6p(12780950471743n^6+97925786606700n^4+297531647010000n^2+3518743761000000),
           p(1345159095200451293561154725n^12+33083931769568719988095485000n^10+52591107015394614602693550000n^8-3601792861738749195338340000000n^6
           -28750555527461688445960500000000n^4-57089481916057216973100000000000n^2+61907788277882125605000000000000+61907788277882125605000000000000n
           +10685911068031014632700000000000n^3-3327499986877866790844100000000n^5-938137908367161577132668000000n^7-70552486831823310670547490000n^9
           -272147669223393846867028200n^11+133231530001027179059168689n^13),
           p(-1345159095200451293561154725n^12-33083931769568719988095485000n^10-52591107015394614602693550000n^8+3601792861738749195338340000000n^6
           +28750555527461688445960500000000n^4+57089481916057216973100000000000n^2-61907788277882125605000000000000+61907788277882125605000000000000n
           +10685911068031014632700000000000n^3-3327499986877866790844100000000n^5-938137908367161577132668000000n^7-70552486831823310670547490000n^9
           -272147669223393846867028200n^11+133231530001027179059168689n^13))

Q.E.D.


3.Example

(p,n,a,m)=(1,6,4,2): 54^6 + 4*(-898)^3 = 16^6 + 4*1762^3
(p,n,a,m)=(1,6,4,3): 64^6 + 4*131602^3 = 486^6 + 4*(-100498)^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