The product of consecutive integers is never a power

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has no solution in integers with k >_ 2, 1 >_ 2 and n >_ 0 . (These restrictions on k, 1 and n will be implicit throughout this paper .) The early literature on this subject can be found in Dickson's history and the somewhat later literature in the paper of Oblath [5] . Rigge [6], and a few months later Erdos [1], proved the conjecture for 1 = 2 . Later these two authors [1] proved that for fixed 1 there are at most finitely many solutions to (1) . In 1940, Erdos and Siegel jointly proved that there is an absolute constant c such that (1) has no solutions with k > c, but this proof was never published . Later Erdos [2] found a different proof ; by improving the method used, we can now completely establish the old conjecture . Thus we shall prove