1.Introduction

We treat Extended Generalised Taxicab Numbers problem, ax113 + bx123 + cx133 = ax213 + bx223 + cx233 = ... = axm13 + bxm23 + cxm33
.

In the same way as Taxicab(4,3,m), we prove that there are always many solutions for any m.

ax113 + bx123 + cx133 = ax213 + bx223 + cx233 = ... = axm13 + bxm23 + cxm33.

We use the parametric solution of ax^3 + by^3 + cz^3 = n below.

See details: ax^3 + by^3 + cz^3 = n 


2.Theorem

There are integers that can be expressed as the sums of cubes in any m of ways.

ax113 + bx123 + cx133 = ax213 + bx223 + cx233 = ... = axm13 + bxm23 + cxm33.
 
condition: a+b=c.
           a,b,p,q,n: arbitrary.


Proof.

We consider the equation ax^3 + by^3 + cz^3 = n.

If a+b=c, then above equation has a parametric solution below.

x = 1/3((2ab^2+a^2b)p^3+(-3a^2b-6ab^2)qp^2+(6ab^2+3a^2b)q^2p+(-a^2b-2ab^2)q^3+a^2n+nb^2+2anb)/((a+b)ab(p-q)^2)
y = 1/3((-2a^2b-ab^2)p^3+(6a^2b+3ab^2)qp^2+(-3ab^2-6a^2b)q^2p+(2a^2b+ab^2)q^3+a^2n+nb^2+2anb)/((a+b)ab(p-q)^2)
z = 1/3((ab^2-a^2b)p^3+(3a^2b-3ab^2)qp^2+(3ab^2-3a^2b)q^2p+(-ab^2+a^2b)q^3-a^2n-nb^2-2anb)/((a+b)ab(p-q)^2).
a,b,p,q,n: arbitrary.

Thus we can obtain distinct rational solutions (x,y,z) for any m as follows.

(x,y,z)=(n11/d11, n12/d11, n13/d11), (n21/d21, n22/d21, n23/d21),...,(nm1/dm1, nm2/dm1, nm3/dm1).

To clear the denominators, multiply the (x,y,z) by d11d21...dm1.

Finally, we obtain the integer solution of ax113 + bx123 + cx133 = ax213 + bx223 + cx233 = ... = axm13 + bxm23 + cxm33.

for any m.



Q.E.D.　


3.Example

m=3, (a,b,c)=(1,2,3), n=36.

(x,y,z)=[11/3, 2/3, -5/3], 
        [101/18, 65/18, -77/18],
        [167/9, 158/9, -161/9].


Finally, we obtain

663 + 2*123 -3*303 = 1013 + 2*653 -3*773 = 3343 + 2*3163 -3*3223.