1.Introduction

We treat Generalised Taxicab Numbers problem, Taxicab(4,3,m) again.
This time, we prove that there are always many solutions for any m.
x114+x124+x134 = x214+x224+x234 = ... = xm14+xm24+xm34
It's pointed out about Taxicab(3,3,m) that there are integers that can be expressed as the sums of three positive cubes in any m of ways in Hardy's book.
See details: 31116963 is expressed as a sum of three positive cubes by 20 ways 

2.Theorem

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

x114+x124+x134 = x214+x224+x234 = ... = xm14+xm24+xm34
m is arbitrary.


Proof.

We consider x1^4+x2^4+x3^4 = N.

Let x1=a, x2=b, x3=a+b.

Hence x1^4+x2^4+x3^4 = a^4+b^4+(a+b)^4 = N.

We use Proth's identity below.

a^4+b^4+(a+b)^4=2(a^2+ab+b^2)^2.

Hence let consider the rational solutions of a^2+ab+b^2=n.

We know that equation a^2+ab+b^2=n has infinitely many rational solutions for some n.

To make the story easy, let n=1.

Of course, the same can be said for other than n = 1.

Quadratic equation a^2+ab+b^2=1 has a parametric solution below.

(a,b)=((k-1)(k+1)/(1+k+k^2), -k(2+k)/(1+k+k^2)).

Thus we can obtain distinct rational solutions (a,b) for any m as follows.

(a,b)=(n11/d11, n12/d11), (n21/d21, n22/d21),...,(nm1/dm1, nm2/dm1).

To clear the denominators, multiply the (a,b) by d11d21...dm1.

We obtain the integer solution of a112 + a11b11 + b112 = a212 + a21b21 + b212 =...= am12 + am1bm1 + bm12 for any m.

Finally, we obtain the integer solution of x114+x124+x134 = x214+x224+x234 = ... = xm14+xm24+xm34 for any m.



Q.E.D.　


3.Example

m=5, a^2+ab+b^2=1

(a,b)=(3/7, -8/7),(8/13, -15/13),(24/31, -35/31),(16/19, -21/19),(33/37, -40/37).

(x,y,z)=(3/7, -8/7, -5/7),(8/13, -15/13, -7/13),(24/31, -35/31, -11/31),(16/19, -21/19, -5/19),(33/37, -40/37, -7/37).

Finally, we obtain 849927^4+ 2266472^4+ 1416545^4 = 1220408^4+ 2288265^4+ 1067857^4
                                                  = 1535352^4+ 2239055^4+ 703703^4
                                                  = 1670032^4+ 2191917^4+ 521885^4
                                                  = 1768767^4+ 2143960^4+ 375193^4