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In this paper, elliptic curves theory is used for solving the quartic Diophantine equation where n ≥ 1, and T i , are rational numbers. We try to transform this quartic to a cubic elliptic curve of positive rank, then get infinitely many integer solutions for the aforementioned Diophantine equation. We solve the above Diophantine equation for some values of n, T i , and obtain infinitely many nontrivial integer solutions for each case. We show among the other things that some numbers can bedoi:10.2307/2373840 fatcat:jlb6tro5xbct7oqtkwyr7ewiny