### Abstract

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 t_{n} ≤ √5n when n > 32. If n = p(2p - 1) and both p and 2p + 1 are primes, then t_{n} = 3p > 3 √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}(1) 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}(2(n) ≤ 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.

Original language | English |
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Pages (from-to) | 137-147 |

Number of pages | 11 |

Journal | Discrete Mathematics |

Volume | 200 |

Issue number | 1-3 |

DOIs | |

Publication status | Published - Apr 6 1999 |

### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics

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## Cite this

*Discrete Mathematics*,

*200*(1-3), 137-147. https://doi.org/10.1016/S0012-365X(98)00332-X