1.Introduction

We treat Generalised Taxicab Numbers problem, Taxicab(5,6,m) again.
In the same way as Taxicab(4,3,m), we prove that there are always many solutions for any m.
x115+x125+x135+x145+x155+x165 = x215+x225+x235+x245+x255+x265 = ... = xm15+xm25+xm35+xm45+xm55+xm65


2.Theorem

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

x115+x125+x135+x145+x155+x165 = x215+x225+x235+x245+x255+x265 = ... = xm15+xm25+xm35+xm45+xm55+xm65
m is arbitrary.


Proof.

We consider x1^5+x2^5+x3^5+x4^5+x5^5+x6^5 = N.

Let x1=t+a, x2=t+b, x3=t+a+b, x4=t-a, x5=t-b, x6=t-a-b.

Hence x1^5+x2^5+x3^5+x4^5+x5^5+x6^5 = 20t(a^2+ab+2t^2+b^2)(a^2+ab+b^2)+6t^5.
t is arbitrary.

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.

For large enough number t, we can obtain the positive integer solution.

Finally, we obtain the positive integer solution of x115+x125+x135+x145+x155+x165 = x215+x225+x235+x245+x255+x265 = ... = xm15+xm25+xm35+xm45+xm55+xm65
for any m.



Q.E.D.　


3.Example

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

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

(x1,x2,x3,x4,x5,x6)=(17/7, 6/7, 9/7, 11/7, 22/7, 19/7),
                    (34/13, 11/13, 19/13, 18/13, 41/13, 33/13),
                    (86/31, 27/31, 51/31, 38/31, 97/31, 73/31),
                    (54/19, 17/19, 33/19, 22/19, 59/19, 43/19),
                    (107/37, 34/37, 67/37, 41/37, 114/37, 81/37).

Finally, we obtain
4816253^5+1699854^5+2549781^5+3116399^5+6232798^5+5382871^5 = 5186734^5+1678061^5+2898469^5+2745918^5+6254591^5+5034183^5
                                                            = 5501678^5+1727271^5+3262623^5+2430974^5+6205381^5+4670029^5
                                                            = 5636358^5+1774409^5+3444441^5+2296294^5+6158243^5+4488211^5
                                                            = 5735093^5+1822366^5+3591133^5+2197559^5+6110286^5+4341519^5.