The difference between the product of n consecutive integers and the nth power of an integer

The difference in tht title is examined in two ways. First, the diophantine equation x(x + 1) … (x + n − 1) = yn + k is considered for integral variables with x ≥ 1, y ≥ 1, and n ≥ 2. We show that for any k, there are only a finite number of x, y, and n satisfying this, and that, in fact, y ≤ k and n < ek. Better restrictions on the solutions are also found. In particular, y and n are both O(k1/3). Second, we look at the value of y that minimizes x(x + 1) … (x + n − 1) − yn and try to find a range for x when a simple formula for such a y exists. We show that the y that minimizes the difference is y = x + [(n − 1)/2] when x is of order at least n2. This is enhanced to show that this formula for y holds when x ≥ (n2 − 1)/(24d) + (13d/10) + O(1/n2) (where d = 1/2 for odd n and d = 1 for even n) and does not hold when x is smaller than this.