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 (p2) > 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(1) 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(2(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.
ASJC Scopus subject areas
- Theoretical Computer Science
- Discrete Mathematics and Combinatorics