ax^3 + by^3 + cz^3 = dxyz

According to Dickson[1], A. Cauchy gave the following formula for finding a new rational
solution to the equation  Ax^3+ By^3+ Cz^3+ Kxyz = 0 from a known solution (a,b,c).
x/(a(Bb^3-Cc^3)) = y/(b(Cc^3-Aa^3)) = z/(c(Aa^3-Bb^3)).

We show diophantine equation ax^3 + by^3 + cz^3 = dxyz has a parametric solution without a known solution.


ax^3 + by^3 + cz^3 = dxyz.......................................................(1)

Let x=pt+1, y=qt+1, z=rt+1,d=a+b+c, then  equation (1) reduces to

(cr^3+bq^3+ap^3-apqr-bpqr-cpqr)t^3
+(-bpq+3ap^2+3cr^2-cpr-apq-bqr+3bq^2-aqr-bpr-cpq-apr-cqr)t^2
+(2ap-ar-cq+2cr-aq-bp-cp+2bq-br)t = 0...........................................(2)

Let r = (2ap-bp-cq+2bq-aq-cp)/(a-2c+b) and t = -(a-2c+b)/(ap-cp+bq-cq).

Then we obtain a very simple parametric solution of equation (1) below.

(x,y,z)=(c-b, a-c, b-a).

a,b,c are arbitrary.



Reference

[1]: Dickson: History of the theory of numbers,  vol II