Question

Is there an efficient algorithm to compute the smallest integer N such that N! is divisible by p^k where p is a relatively small prime number and k, a very large integer. In other words,

factorial(N) mod p^k == 0

If, given N and p, I wanted to find how many times p divides into N!, I would use the well-known formula

k = Sum(floor(N/p^i) for i=1,2,...

I’ve done brute force searches for small values of k but that approach breaks down very quickly as k increases and there doesn’t appear to be a pattern that I can extrapolate to larger values.

Edited 6/13/2011

Using suggestions proposed by Fiver and Hammar, I used a quasi-binary search to solve the problem but not quite in the manner they suggested. Using a truncated version of the second formula above, I computed an upper bound on N as the product of k and p (using just the first term). I used 1 as the lower bound. Using the classic binary search algorithm, I computed the midpoint between these two values and calculated what k would be using this midpoint value as N in the second formula, this time with all the terms being used.

If the computed k was too small, I adjusted the lower bound and repeated. Too big, I first tested to see if k computed at midpoint-1 was smaller than the desired k. If so, midpoint was returned as the closest N. Otherwise, I adjusted the highpoint and repeated.

If the computed k were equal, I tested whether the value at midpoint-1 was equal to the value at midpoint. If so, I adjusted the highpoint to be the midpoint and repeated. If midpoint-1 was less than the desired k, the midpoint was returned as the desired answer.

Even with very large values for k (10 or more digits), this approach works O(n log(n)) speeds.

Answer 1

Using the formula you mentioned, the sequence of k values given fixed p and N = 1,2... is non-decreasing. This means you can use a variant of binary search to find N given the desired k.

• Start with N = 1, and calculate k.
• Double N until k is greater or equal than your desired k to get an upper bound.
• Do a binary search on the remaining interval to find your k.