1.Introduction

We show that x^5+y^5=cz^2 has infinitely many integer solutions
                 
                 
2.Theorem
     
Diophantine equation x^5+y^5=cz^2 has infinitely many integer solutions.

condition: 
p,q are arbitrary.
c=p^5+q^5

 
Proof.

x^5+y^5=cz^2...........................................................................................................(1)

Let x=t+p, y=-t+q, the equation (1) becomes to below equation.

cz^2 = (5p+5q)t^4+(-10q^2+10p^2)t^3+(10q^3+10p^3)t^2+(5p^4-5q^4)t+p^5+q^5..............................................(2)

Let c=p^5+q^5, U=t, V=cz, then we obtain

V^2 = (p^5+q^5)(5p+5q)U^4+(p^5+q^5)(-10q^2+10p^2)U^3+(p^5+q^5)(10q^3+10p^3)U^2+(p^5+q^5)(5p^4-5q^4)U+(p^5+q^5)^2.......(3)
    
Since quartic equation (3) has a rational point Q(U,V)=(0,p^5+q^5), then this quartic equation is birationally equivalent to an elliptic curve below.

Y^2+(5p^4-5q^4)YX+(-20p^10q^2+20p^12-40p^5q^7+40p^7q^5-20q^12+20q^10p^2)Y
= X^3+(15/4q^8+10q^5p^3+25/2p^4q^4+10p^5q^3+15/4p^8)X^2
+(-20p^16-60p^11q^5-60p^6q^10-20p^15q-60p^10q^6-60p^5q^11-20q^15p-20q^16)X
-75q^24-75q^23p-200q^21p^3-450q^20p^4-675p^5q^19-425p^6q^18-675p^8q^16-1425p^9q^15
-1575p^10q^14-825p^11q^13-825p^13q^11-1575p^14q^10-1425p^15q^9-675p^16q^8-425p^18q^6
-675p^19q^5-450p^20q^4-200p^21q^3-75p^23q-75p^24

Transformation is given, 

U = 1/2(15q^13+40p^3q^10+50p^4q^9+55p^5q^8+4q^5X+55p^8q^5+50p^9q^4+40p^10q^3+4p^5X+15p^13)/Y

V = 1/8(160p^20Xq-160p^15Yq^2+160q^20Xp+160q^15Yp^2+3600q^25p^4+600q^28p+1600q^26p^3+1600p^26q^3
    +7000p^8q^21+15000p^9q^20+4000p^6q^23+12000p^13q^16+18000p^10q^19+10000p^18q^11+4000p^23q^6+600p^28q
    +24000p^15q^14+15000p^20q^9+3600p^25q^4+6000p^5q^24+8q^5X^3+90q^13X^2-10q^17Y+385Xq^21+24000p^14q^15
    +10000p^11q^18+18000p^19q^10+6000p^24q^5+12000p^16q^13+7000p^21q^8+600q^29+600p^29+90p^13X^2+2240p^15Xq^6
    +2240p^6Xq^15+2065p^5Xq^16+1200Xp^3q^18+8p^5X^3-400q^3p^14Y+1200q^3p^18X+240q^3p^10X^2+1500q^4p^17X
    +300q^4p^9X^2-350q^4p^13Y+2065q^5p^16X-70q^5p^12Y+330q^5p^8X^2+4000q^7p^14X-80q^7p^10Y+7350q^8p^13X
    -50q^8p^9Y+330q^8X^2p^5+50q^9p^8Y+300q^9p^4X^2+5500q^9p^12X+240q^10X^2p^3+80q^10p^7Y+3760q^10p^11X
    +70q^12p^5Y+5500q^12p^9X+7350q^13p^8X+350q^13p^4Y+4000q^14p^7X+400q^14p^3Y+1500q^17p^4X+10p^17Y+385p^21X
    +3760p^10Xq^11)/(Y^2)

The point corresponding to point Q is P(X,Y)=(-15/4q^8-10q^5p^3-25/2p^4q^4-10p^5q^3-15/4p^8,
5/4q^12-20q^10p^2-50p^3q^9-175/4q^8p^4-10p^5q^7+10p^7q^5+175/4p^8q^4+50p^9q^3+20p^10q^2-5/4p^12).

Hence we get 2Q(U,V)=(8(q^4-pq^3+p^2q^2-p^3q+p^4)(-p+q)(p^4-6p^3q+6p^2q^2-6pq^3+q^4)
/(19q^8-88pq^7+172p^2q^6-216q^5p^3+210p^4q^4-216p^5q^3+172p^6q^2-88p^7q+19p^8),
(p+q)(q^4-pq^3+p^2q^2-p^3q+p^4)(101q^16-464q^15p+1640q^14p^2-5040q^13p^3+11340p^4q^12-16912p^5q^11
+18648p^6q^10-17520q^9p^7+16670p^8q^8-17520p^9q^7+18648p^10q^6
-16912p^11q^5+11340q^4p^12-5040p^13q^3+1640p^14q^2-464p^15q+101p^16)
/((19q^8-88pq^7+172p^2q^6-216q^5p^3+210p^4q^4-216p^5q^3+172p^6q^2-88p^7q+19p^8)^2) ).

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 = (19q^8-88pq^7+172p^2q^6-216q^5p^3+210p^4q^4-216p^5q^3+172p^6q^2-88p^7q+19p^8)
    (8q^9+11p^9-24p^8q-102q^4p^5+96q^5p^4-84q^6p^3+72q^7p^2-45q^8p+40p^6q^3+12p^7q^2)
    
y = (19q^8-88pq^7+172p^2q^6-216q^5p^3+210p^4q^4-216p^5q^3+172p^6q^2-88p^7q+19p^8)
    (11q^9+8p^9-45p^8q+96q^4p^5-102q^5p^4+40q^6p^3+12q^7p^2-24q^8p-84p^6q^3+72p^7q^2)
    
z = (101q^16-464q^15p+1640q^14p^2-5040q^13p^3+11340p^4q^12-16912p^5q^11+18648p^6q^10-17520q^9p^7+16670p^8q^8-17520p^9q^7+18648p^10q^6-16912p^11q^5+11340q^4p^12-5040p^13q^3+1640p^14q^2-464p^15q+101p^16)
    (19q^8-88pq^7+172p^2q^6-216q^5p^3+210p^4q^4-216p^5q^3+172p^6q^2-88p^7q+19p^8)^3
       
Q.E.D.