There are many solutions for A1.Introduction_{1}^{6}+ A_{2}^{6}+ A_{3}^{6}+ A_{4}^{6}= B_{1}^{6}+ B_{2}^{6}+ B_{3}^{6}+ B_{4}^{6}. I show that there are infinitely many solutions for 6.4.4 by different from Crussol's method, and describe a method of generating solutions. [1]:Dickson: History of the Theory of Numbers V2There are infinitely many solutions for A2. Theorem_{1}^{6}+ A_{2}^{6}+ A_{3}^{6}+ A_{4}^{6}= B_{1}^{6}+ B_{2}^{6}+ B_{3}^{6}+ B_{4}^{6}. A_{1}^{6}+ A_{2}^{6}+ A_{3}^{6}+ A_{4}^{6}= B_{1}^{6}+ B_{2}^{6}+ B_{3}^{6}+ B_{4}^{6}.................(1) A_{1}=(3q+p)d+(q-p)c A_{2}=(q+p)d+(q-3p)c A_{3}=(p-q)d+(3q+p)c A_{4}=(3p-q)d+(q+p)c B_{1}=(3q-p)d+(q+3p)c B_{2}=(q-p)d+(q+p)c B_{3}=(q+p)d+(p-q)c B_{4}=(q+3p)d+(p-3q)c Proof. We use above Crussol's parameter.([1]:Dickson) A_{1}^{6}+ A_{2}^{6}+ A_{3}^{6}+ A_{4}^{6}-( B_{1}^{6}+ B_{2}^{6}+ B_{3}^{6}+ B_{4}^{6}) =480qp(p^{2}+q^{2})(-2c+d)(2d+c)(c^{2}+d^{2})(3d^{2}q^{2}-3d^{2}p^{2}-4dp^{2}c+4dq^{2}c+6dqcp-3c^{2}q^{2}+3c^{2}p^{2}) (3d^{2}q^{2}-3d^{2}p^{2}-4dp^{2}c+4dq^{2}c+6dqcp-3c^{2}q^{2}+3c^{2}p^{2})=(3q^{2}-3p^{2})d^{2}+(-4p^{2}+4q^{2}+6qp)cd+(-3q^{2}+3p^{2})c^{2}= 0 We must find rational value (c,d) for above equation. Discriminant=13q^{4}-17q^{2}p^{2}+12q^{3}p+13p^{4}-12p^{3}q=y^{2}...................(2) So, we must find rational numbers p,q,y. U=q/p V=y/p^{2}V^{2}= 13U^{4}+12U^{3}-17U^{2}-12U+13.....................................(3) Using APECS program by Ian Connell,Weierstrass form is Y^{2}= X^{3}+X^{2}-916X+9920...........................................(4) U = (X-128-Y)/(-Y+7X-32) V = (-864Y+144X^{2}+2844X-103488+3X^{3})/(-Y+7X-32)^{2}.................(5) X = (6V+8-22U+32U^{2})/(U^{2}-2U+1) Y = (-6V+120-210U+6U^{2}+42VU+192U^{3})/(U^{3}-3U^{2}+3U-1)................(6) Point P(1/13,45/13) satisfys (3). Rational point Q(X,Y) on the curve (4) corresponding to the values U=1/13,V=45/13 is X=32,Y=-120 by using (6). So, we get the relation of the curve (3) and the curve (4). Point P(1/13,45/13) on the curve (3) <=======> Point Q(32,-120) on the curve (4) We obtain 2Q=(329/16, 909/64) on the curve (4) using APECS. As this point on the curve (4) does not have intger coordinates, there are infinitely many rational points on the curve (4) by Nagell-Lutz theorem. Point 2P=(-173/139,60051/19321) is given by 2Q using (5). We can get a new solution by point 2P=(-173/139,60051/19321). 13806^{6}+ 39457^{6}+ 31924^{6}+ 13817^{6}= 29423^{6}+ 3772^{6}+21879^{6}+ 39986^{6}We can obtain infinitely many integer solutions for (1) by apllying the group law. Q.E.D.

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