There are many solutions for A1.Introduction_{1}^{6}+ A_{2}^{6}+ A_{3}^{6}+ A_{4}^{6}+ A_{5}^{6}+ A_{6}^{6}= B_{1}^{6}+ B_{2}^{6}+ B_{3}^{6}+ B_{4}^{6}+ B_{5}^{6}+ B_{6}^{6}. I show that there are infinitely many solutions for 6.6.6,and describe a method of generating solutions.There are infinitely many solutions for A2. Theorem_{1}^{6}+ A_{2}^{6}+ A_{3}^{6}+ A_{4}^{6}+ A_{5}^{6}+ A_{6}^{6}= B_{1}^{6}+ B_{2}^{6}+ B_{3}^{6}+ B_{4}^{6}+ B_{5}^{6}+ B_{6}^{6}. A_{1}^{6}+ A_{2}^{6}+ A_{3}^{6}+ A_{4}^{6}+ A_{5}^{6}+ A_{6}^{6}= B_{1}^{6}+ B_{2}^{6}+ B_{3}^{6}+ B_{4}^{6}+ B_{5}^{6}+ B_{6}^{6}..................(1) A_{1}=qd-(p-q)c A_{2}=(p+q)d-pc A_{3}=(p-q)d+qc A_{4}=pd+(p+q)c A_{5}=(p+2q)d-(2p-q)c A_{6}=(2p-q)d+(p+2q)c B_{1}=qd+pc B_{2}=(p-q)d-(p+q)c B_{3}=(p+q)d+(p-q)c B_{4}=pd-qc B_{5}=(2p-q)d-(p+2q)c B_{6}=(p+2q)d+(2p-q)c Proof. A_{1}^{6}+ A_{2}^{6}+ A_{3}^{6}+ A_{4}^{6}+ A_{5}^{6}+ A_{6}^{6}-( B_{1}^{6}+ B_{2}^{6}+ B_{3}^{6}+ B_{4}^{6}+ B_{5}^{6}+ B_{6}^{6}) =15(c^{2}+d^{2})dc(-2p+q)(p+2q)*(p^{2}+q^{2})(-12q^{2}c^{2}+12q^{2}d^{2}+34pd^{2}q-3qdpc-34pc^2q+12p^{2}c^{2}-12p^{2}d^{2}) (-12q^{2}c^{2}+12q^{2}d^{2}+34pd^{2}q-3qdpc-34pc^{2}q+12p^{2}c^{2}-12p^{2}d^{2}) =(34pq+12q^{2}-12p^{2})d^{2}-3qdpc+(-12q^{2}+12p^{2}-34pq)c^{2}= 0 We must find rational value (c,d) for above equation. Discriminant=3481p^{2}q^{2}+3264pq^{3}-3264p^{3}q+576q^{4}+576p^{4}=y^{2}........................(2) So, we must find rational numbers p,q,y. U=q/p V=y/p^{2}V^{2}= 576U^{4}+3264U^{3}+3481U^{2}-3264U+576..........................................(3) Using APECS program by Ian Connell,Weierstrass form is Y^{2}+XY = X^{3}-1001245X+ 385535537..............................................(4) U = (48X-27636)/(2Y+69X-39304) V = (14688Y-165816X^{2}+96099228X-18682009464+96X^{3})/((2Y+69X-39304)^{2})..........(5) X = (12V+288-816U+290U^{2})/U^{2}Y = (288V+6912-29520U+21294U^{2}-414VU+9647U^{3})/U^{3}..............................(6) Point P(0,-24) satisfys (3). Rational point Q(X,Y) on the curve (4) corresponding to the values U=0,V=-24 is X=2303/4,Y=-2915/8 by using (6). So, we get the relation of the curve (3) and the curve (4). Point P(0,-24) on the curve (3) <=======> Point Q(2303/4,-2915/8) on the curve (4) We obtain 2Q=(166193/256, -13782079/4096) 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=(-128/45,16712/675) is given by 2Q using (5). We can get the new solution by point 2P=(-128/45,16712/675). 4433^{6}+ 686^{6}+ 13185^{6}+ 6064^{6}+ 3747^{6}+ 19249^{6}= 7121^{6}+ 2002^{6}+ 12240^{6}+ 5119^{6}+ 3117^{6}+ 19361^{6}We can obtain infinitely many integer solutions for (1) by apllying the group law. Q.E.D.

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