We treat Generalised Taxicab Numbers problem, Taxicab(4,3,m) again. This time, we prove that there are always many solutions for any m. x1.Introduction_{11}^{4}+x_{12}^{4}+x_{13}^{4}= x_{21}^{4}+x_{22}^{4}+x_{23}^{4}= ... = x_{m1}^{4}+x_{m2}^{4}+x_{m3}^{4}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 waysProof. 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)=(n2.TheoremThere are integers that can be expressed as the sums of three fourth powers in any m of ways. x_{11}^{4}+x_{12}^{4}+x_{13}^{4}= x_{21}^{4}+x_{22}^{4}+x_{23}^{4}= ... = x_{m1}^{4}+x_{m2}^{4}+x_{m3}^{4}m is arbitrary._{11}/d_{11}, n_{12}/d_{11}), (n_{21}/d_{21}, n_{22}/d_{21}),...,(n_{m1}/d_{m1}, n_{m2}/d_{m1}). To clear the denominators, multiply the (a,b) by d_{11}d_{21}...d_{m1}. We obtain the integer solution of a_{11}^{2}+ a_{11}b_{11}+ b_{11}^{2}= a_{21}^{2}+ a_{21}b_{21}+ b_{21}^{2}=...= a_{m1}^{2}+ a_{m1}b_{m1}+ b_{m1}^{2}for any m. Finally, we obtain the integer solution of x_{11}^{4}+x_{12}^{4}+x_{13}^{4}= x_{21}^{4}+x_{22}^{4}+x_{23}^{4}= ... = x_{m1}^{4}+x_{m2}^{4}+x_{m3}^{4}for any m. Q.E.D.@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+ 375193^43.Example

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