### 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 |

Publication status | Published - Apr 6 1999 |

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### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

*Discrete Mathematics*,

*200*(1-3), 137-147.

**Subsets of an interval whose product is a power.** / Erdős, P.; Malouf, Janice L.; Selfridge, J. L.; Szekeres, Esther.

Research output: Contribution to journal › Article

*Discrete Mathematics*, vol. 200, no. 1-3, pp. 137-147.

}

TY - JOUR

T1 - Subsets of an interval whose product is a power

AU - Erdős, P.

AU - Malouf, Janice L.

AU - Selfridge, J. L.

AU - Szekeres, Esther

PY - 1999/4/6

Y1 - 1999/4/6

N2 - 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 2(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.

AB - 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 2(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.

UR - http://www.scopus.com/inward/record.url?scp=0347040224&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0347040224&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0347040224

VL - 200

SP - 137

EP - 147

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 1-3

ER -