Subsets of an interval whose product is a power Original Research Article

We form squares from the product of integers in a short interval [n, n + tn], where we include n in the product. If p is prime, p|n, and (2p) > n, we prove that p is the minimum tn. If no such prime exists, we prove tn ⩽ √5n when n > 32. If n = p(2p − 1) and both p and 2p − 1 are primes, then tn = 3p > 3 √n/2. For n(n + u) a square > n2, we conjecture that a and b exist where n < a < b < n + u and nab is a square (except n = 8 and n = 392). Let g2(n) be minimal such that a square can be formed as the product of distinct integers from [n, g2(n)] so that no pair of consecutive integers is omitted. We prove that g2(n) ⩽ 3n − 3, and list or conjecture the values of g2(n) for all n. We describe the generalization to kth powers and conjecture the values for large n.