1.Introduction

We show that X^5 + aY^5 + aZ^5 = W^2 has infinitely many integer solutions.

Related topics:  ax^5 + by^5 + cz^5 = w^2. 
                 ax^5 + by^5 + cz^5 = w^2 Part2. 
                 
                 
2.Theorem
     
Equation X^5 + aY^5 + aZ^5 = W^2 has infinitely many integer solutions.

condition: a=-p+1, p is arbitrary.


 
Proof.

X^5 + aY^5 + aZ^5 = W^2.........................................................(1)

Substitute X=p, Y=t-p, Z=-t to equation (1), we obtain

W^2 = -5apt^4+10ap^2t^3-10ap^3t^2+5ap^4t+p^5-ap^5...............................(2)

Let a=-p+1, U=t, and V=W, then we obtain equation (3).

V^2 = (-5p+5p^2)U^4+(-10p^3+10p^2)U^3+(10p^4-10p^3)U^2+(-5p^5+5p^4)U+p^6........(3)


Since quartic equation (3) has a rational solution Q(U,V)=(0,p^3),
this quartic equation (3) is birationally equivalent to an elliptic curve below.

Y^2+(-5p^2+5p)YX+(-20p^6+20p^5)Y
= X^3+(15/4p^4+5/2p^3-25/4p^2)X^2+(20p^7-20p^8)X+25p^11-75p^12+175p^10-125p^9

Transformation is given, 
U = 1/2(4p^3X+15p^7+10p^6-25p^5)/Y
V = 1/8(625p^7X-500p^8X+90p^7X^2+60p^6X^2+8p^3X^3-150p^5X^2+385p^11X-200p^14+600p^15-1400p^13+1000p^12+140p^10X-650p^9X-10p^9Y+110p^8Y-350p^7Y+250p^6Y)/(Y^2)
X = (2p^3V+2p^6-5Up^5+5Up^4)/(U^2)
Y = 1/2(8p^6V+8p^9-20Up^8+20Up^7+15U^2p^7+10U^2p^6-25U^2p^5)/(U^3).

The point corresponding to point Q is P(X,Y)=(-15/4p^4-5/2p^3+25/4p^2, 5/4p^6-55/4p^5+175/4p^4-125/4p^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

X = p(19p^2-10p-25)^2
Y = -p(11p^2+30p-25)(19p^2-10p-25)
Z = -8p^2(p-5)(19p^2-10p-25)
W = p^3(19p^2-10p-25)^3(-1875+2500p-1450p^2+980p^3+101p^4)
p is arbitrary.

Case : m=3

X = p(181p^6-12210p^5-74625p^4+164500p^3-108125p^2+18750p+15625)^2
Y = p(11p^2+30p-25)(3p^2+70p-25)(19p^2-10p-25)(181p^6-12210p^5-74625p^4+164500p^3-108125p^2+18750p+15625)
Z = -8(p-5)p^2(-1875+2500p-1450p^2+980p^3+101p^4)(181p^6-12210p^5-74625p^4+164500p^3-108125p^2+18750p+15625)
W = p^3(1220703125-4882812500p+12675781250p^2-32507812500p^3+44176171875p^4-18570625000p^5-11322562500p^6+12498875000p^7-4048103125p^8+521613500p^9+225617250p^10+28761500p^11+1169341p^12)
       (181p^6-12210p^5-74625p^4+164500p^3-108125p^2+18750p+15625)^3

Q.E.D.