1.Introduction



Ryley's Theorem[1]: "Any non-zero rational number m is the sum of three rational cubes
in an infinite number of non-trivial ways." 

(-81d^6a^2+27d^9+45a^4d^3+a^6)^3 + (27a^4d^3+81d^6a^2-27d^9-a^6)^3 + (-36a^4d^3+108d^6a^2)^3
= a(6a^5d+36a^3d^4+54ad^7)^3

By Tito Piezas[2],it seems that X^3+Y^3+Z^3 = m has a follwing parametric solution.

Yuri Manin :
(m^3-3^6n^9)^3 + (-m^3+3^5mn^6+3^6n^9)^3 + (3^3m^2n^3+3^5mn^6)^3 = m(3^2m^2n^2 +3^4mn^5+3^6n^8)^3

Robert Israel :
(27m^3-n^9)^3 + (-27m^3+9mn^6+n^9)^3 + (27m^2n^3+9mn^6)^3 = m(27m^2n^2 +9mn^5+3n^8)^3 

H.W.Richmond[3] :
(s^9+27n^3)^3 + (9ns^6-s^9-27n^3)^3 + (9s^3n(s^3-3n))^3 = n(3(s^6-3ns^3+9n^2)s^2)^3

All the identities of Manin,Israel, and Richmond are the same.


I found a new parametric solution of X^3+Y^3+Z^3 = m by using a known solution.


[1] L.E. Dickson: History of the theory of numbers,vol.2
[2] Tito Piezas:http://sites.google.com/site/tpiezas/001b
[3] L.J.Mordell: Diophantine equations



2.Theorem
         
     
　　Any non-zero rational number m is the sum of three rational cubes
    in an infinite number of non-trivial ways,

    X^3+Y^3+Z^3 = mW^3.

    X = -(m-9n^3)(54m^5n^3+1215m^4n^6+11664m^3n^9+59049m^2n^12+m^6+177147mn^15+531441n^18)(m^2+9mn^3+81n^6)

    Y = (54m^5n^3+1215m^4n^6+11664m^3n^9+59049m^2n^12+m^6+177147mn^15+531441n^18)(m^3-243mn^6-729n^9)

    Z = 27(m+9n^3)(27m^5n^3+243m^4n^6+3645m^3n^9+59049m^2n^12+2m^6+354294mn^15+1062882n^18)mn^3

    W = -9n^2(m^2+9mn^3+81n^6)(m^6-27m^5n^3-972m^4n^6-8019m^3n^9+177147mn^15+531441n^18)


     
Proof.

X^3+Y^3+Z^3 = mW^3.....................................................(1)

Let {x0,y0,z0} is a known solution of X^3+Y^3+Z^3 = m.

X=x0x, Y=y0x, Z=z0x+a, W=x+b...........................................(2)

Substitute (2) to (1), and simplifying (1),we obtain

(y0^3+z0^3+x0^3-m)x^3+(3az0^2-3mb)x^2+(3a^2z0-3mb^2)x+a^3-mb^3


Choosing a=m and b=z0^2,then we obtain

x=-1/3(z0^3+m)/z0

We use a following parametric solution as a known solution.

x0=(m^3-729n^9)/(9m^2n^2+81mn^5+729n^8)   

y0=(-m^3+243mn^6+729n^9)/(9m^2n^2+81mn^5+729n^8)

z0=(27m^2n^3+243mn^6)/(9m^2n^2+81mn^5+729n^8)

Substitute x0 , y0, z0, x, a, and b to (2),and obtain a new parametric solution. 
    

X = -(m-9n^3)(54m^5n^3+1215m^4n^6+11664m^3n^9+59049m^2n^12+m^6+177147mn^15+531441n^18)(m^2+9mn^3+81n^6)

Y = (54m^5n^3+1215m^4n^6+11664m^3n^9+59049m^2n^12+m^6+177147mn^15+531441n^18)(m^3-243mn^6-729n^9)

Z = 27(m+9n^3)(27m^5n^3+243m^4n^6+3645m^3n^9+59049m^2n^12+2m^6+354294mn^15+1062882n^18)mn^3

W = -9n^2(m^2+9mn^3+81n^6)*(m^6-27m^5n^3-972m^4n^6-8019m^3n^9+177147mn^15+531441n^18)



       


   
Q.E.D.　
 
　　                  
       
3.Example


m=1..10
n=1

(   -6244568/6296139)^3 +(     757934441/572948649)^3 +(    -44404260/63660961)^3 = 1
(   -1236475/1035297)^3 +(    1492425325/746449137)^3 +(   -134752596/82938793)^3 = 2
(         -5546/3135)^3 +(            146969/40755)^3 +(          -45816/13585)^3 = 3
(  -16508705/4089861)^3 +(    5404950017/543951513)^3 +(   -585962520/60439057)^3 = 4
(   21180604/2353041)^3 +(   -9631879669/355309191)^3 +(   1057077420/39478799)^3 = 5
(         11359/6423)^3 +(          -829207/122037)^3 +(          276540/40679)^3 = 6
(  25216130/32840631)^3 +( -26313031655/6338241783)^3 +(  3010259616/704249087)^3 = 7
(  18707977/60849135)^3 +(-40427938297/13204262295)^3 +( 4874360496/1467140255)^3 = 8
(                  0)^3 +(                   -17/7)^3 +(                  20/7)^3 = 9
(-38421811/154224801)^3 +(-82952689949/41794921071)^3 +(12132861540/4643880119)^3 = 10