1.Introduction


We show a parametric solution for x^m + ky^3 = z^m + kw^3.
Hence we can show x^m + ky^3 = z^m + kw^3 has infinitely many rational solutions for any rational number k.

2.Theorem
      
 
Diophantine equation x^m + ky^3 = z^m + kw^3 has infinitely many rational solutions for any rational number k.

(x,y,z,w)=( a^3/k^(β), (-2b^ma^(3m)+b^(4m))/((b^(3m)+a^(3m))k^(α)), b^3/k^(β), (-2a^mb^(3m)+a^(4m))/((b^(3m)+a^(3m))k^(α))  ).
    
     a, b, k, m: arbitrary
     gcd(m,3)=1
     3α-mβ=1

     
Proof.

x^m + ky^3 = z^m + kw^3.....................................................(1)

As A.S. Janfada et al.[1] mentioned to X^m + Y^3 = Z^m + W^3 below.
Let m,3 be coprime integers.
An equation x^m + ky^3 = z^m + kw^3 can be reduced to X^m + Y^3 = Z^m + W^3.
The positive integers α,β exist such that 3α-mβ=1.
Multiplying both side of (1) by k^(βm), we obtain the result. 

Hence first, we obtain a parametric solution of X^m + Y^3 = Z^m + W^3.

X^m + Y^3 = Z^m + W^3.......................................................(2)

Substitute X=a^3, Y=t+b^m, Z=b^3, W=pt+a^m to equation (2).

Let p = b^(2m)/(a^(2m), t = -3b^ma^(3m)/(b^(3m)+a^(3m)) then we obtain a parametric solution

(X,Y,Z,W)=( a^3, (-2b^ma^(3m)+b^(4m))/(b^(3m)+a^(3m)), b^3, (-2a^mb^(3m)+a^(4m))/(b^(3m)+a^(3m)) ).

Next, we obtain the integers α,β such that 3α-mβ=1.

Hence we obtain a parametric solution for x^m + ky^3 = z^m + kw^3 below.

(x,y,z,w)=(X/k^(β), Y/k^(α), Z/k^(β), W/k^(α)).
 
Q.E.D.
　


3.Example

x^7 + ky^3 = z^7 + kw^3.

(α,β)=(5,2)

(x,y,z,w)=(a^3/(k^2), -b^7(2a^21-b^21)/((a^21+b^21)k^5), b^3/(k^2), a^7(-2b^21+a^21)/((a^21+b^21)k^5) )

k=2: (x,y,z,w)=(2, -1398101/22369632, 1/4, 2796200/699051)
k=3: (x,y,z,w)=(8/9, -1398101/169869393, 1/9, 89478400/169869393)
k=4: (x,y,z,w)=(1/2, -1398101/715828224, 1/16, 349525/2796204)
k=5: (x,y,z,w)=(8/25, -1398101/2184534375, 1/25, 3579136/87381375)
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4.Reference

[1]:  N. Yousefnejad, H. Shabani-Solt, and A. S. Janfada, On theDiophantine equation x^5 + ky^3 = u^5 + kv^3 , Mathematica, Vol.60 (83), No. 1 (2018), 95–99.