1.Introduction

By Tito Piezas[1], it seems that ax^3+by^3 = t^2 has a follwing parametric solution.

x = 4p(ap^3-bq^3)
y = q(8ap^3+bq^3)
t = -b^2q^6+20ap^3bq^3+8a^2p^6

We show the parametric solution for ax^3+by^3 = t^k for k>1.


2.Theorem
      
 a,b,k: arbitrary

 condition: 
    
     k>1
     n=lcm(3,k)
     ap^3 + bq^3 = r^k.
     
There is a parametric solution of ax^3 + by^3 = t^k,

      x = p(2bq^3+ap^3)(ap^3-bq^3)^(n/3-1)

      y = -q(2ap^3+bq^3)(ap^3-bq^3)^(n/3-1)

      t = r(ap^3-bq^3)^(n/k).


     
Proof.

Let ap^3 + bq^3 = r^k and n=lcm(3,k).

Set x = m+p, y= sm+q, t = r.......................................................(1)

a(m+p)^3 + b(sm+q)^3 - r^k = (a+bs^3)m^3+(3ap+3bqs^2)m^2+(3ap^2+3bq^2s)m..........(2)

Set s=-ap^2/(bq^2), then we obtain m = 3pbq^3/(ap^3-bq^3).

Substitute s and m to (1), and obtain a parametric solution

 
Q.E.D.
　

3.Examples


Case. k=2

Using a known solution (p,q,r)=(4m(am^3-bn^3), n(8am^3+bn^3), -b^2n^6+20am^3bn^3+8a^2m^6), we obtain new solution below.

x = 8m(am^3-bn^3)(416a^3m^9bn^3+288b^2n^6a^2m^6-8b^3n^9am^3+b^4n^12+32a^4m^12)(b^2n^6+8a^2m^6)(-b^2n^6-88am^3bn^3+8a^2m^6)
y = -n(8am^3+bn^3)(128a^4m^12+128a^3m^9bn^3+576b^2n^6a^2m^6-104b^3n^9am^3+b^4n^12)(b^2n^6+8a^2m^6)(-b^2n^6-88am^3bn^3+8a^2m^6)
t = (-b^2n^6+20am^3bn^3+8a^2m^6)(b^2n^6+8a^2m^6)^3(-b^2n^6-88am^3bn^3+8a^2m^6)^3
a,b,m,n: arbitrary

Case. k=3

x = p(2bq^3+ap^3)
y = -q(2ap^3+bq^3)
t = r(ap^3-bq^3)

Case. k=4

x = p(2bq^3+ap^3)(ap^3-bq^3)^3
y = -q(2ap^3+bq^3)(ap^3-bq^3)^3
t = r(ap^3-bq^3)^3




4.Reference

[1].Tito Piezas:http://sites.google.com/site/tpiezas/012