The difference between the product of n consecutive integers and the n(th)power of an integer

The difference in tht title is examined in two ways. First, the diophantine equation x(x + l)...(x + n - 1) = y(n) + k is considered for integral vari ables with x greater than or equal to 1, y greater than or equal to 1, and n greater than or equal to 2. We show that for any k, there are only a fini te number of x, y, and n satisfying this, and that, in fact, y less than or equal to \k\ and n < e\k\. Better restrictions on the solutions are also f ound. In particular, y and n are both O(\k\(1/3)). Second, we look at the v alue of y that minimizes \x(x + 1)...(x + n - 1) - y(n)\ and try to find a range for x when a simple formula for such a y exists. We show that the y t hat minimizes the difference is y = x + [(n - 1)/2] when x is of order at l east n(2). This is enhanced to show that this formula for y holds when x gr eater than or equal to (n(2) - 1)/(24d) + (13d/10) + O(1/n(2)) (where d = 1 /2 for odd n and d = 1 for even n) and does not hold when x is smaller than this. (C) 2000 Elsevier Science Ltd. All rights reserved.