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We obtain explicit forms of the current best known asymptotic upper bounds for gaps between cubefree integers. In particular, we show that the interval \((x, x + 5x^{1/7}\log x]\) contains a cubefree integer for any \(x \ge 2\) . The constant 5 can be improved further, if x is assumed to be larger than a very large constant.

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Explicit Bounds for Large Gaps Between Cubefree Integers

  • 会议论文

  • 发表日期：

    2025年

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  • Angel Kumchev,
  • Wade McCormick,
  • Nathan McNew,
  • Ariana Park,
  • Russell Scherr,
  • Willow Ziehr

摘要

We obtain explicit forms of the current best known asymptotic upper bounds for gaps between cubefree integers. In particular, we show that the interval \((x, x + 5x^{1/7}\log x]\) contains a cubefree integer for any \(x \ge 2\) . The constant 5 can be improved further, if x is assumed to be larger than a very large constant.