Abstract
AbstractThe Diophantine equation X(X + 1)(X + 2)(X + 3) = 15Y(Y + 1)(Y + 2)(Y + 3) still remain open. Using recurrence sequence, Maple software, Pell equation and quadratic residue, this paper proved it has only two positive integer solutions, i.e., (X,Y) = (3,1), (25,12). And the Diophantine equation X 2 −15(X 2 + 3X + 1)2= −14 which are ( ± X,Y) = (1,−1), (1,−2), (1,−3), (1,0), (19, −4), (19, 1), (701, −15), (701,12).
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