Subsets of an interval whose product is a power

We form squares from the product of integers in a short interval [n, n + t(n)], where we include n in the product. If p is prime, p\n, and ((p)(2)) > n, we prove that p is the minimum t(n). If no such prime exists, we prove tn less than or equal to root 5n when n > 32. If n = p(2p - 1) and both p and 2p + 1 are primes, then t(n) = 3p >3 root n\2. For n(n + u) a square > n(2), 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 g(2)(n) be minimal such that a square can be formed as the product of distinct integers from [n, g(2)(n)] so that no pair of consecutive integers is omitted. We prove that g(2)(n) less than or equal to 3n - 3, and list or conjecture the values of g(2)(n) for all n. We describe the generalization to kth powers and conjecture the values for large n. (C) 1999 Elsevier Science B.V. All rights reserved.