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<p>We prove that there exist only three pentagonal numbers that can be written as the product of three consecutive integers. We also show that there does not exist any pentagonal number that is a perfect cube. To prove these results, we used the arithmetic of elliptic curves and fundamental units of certain real quadratic field to find the integral solutions of a resulting elliptic curve.</p>

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On Certain Pentagonal Numbers

  • 期刊论文

  • 发表日期：

    2025年8月16日

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  • Amrutha C,
  • Leena Mondal

摘要

We prove that there exist only three pentagonal numbers that can be written as the product of three consecutive integers. We also show that there does not exist any pentagonal number that is a perfect cube. To prove these results, we used the arithmetic of elliptic curves and fundamental units of certain real quadratic field to find the integral solutions of a resulting elliptic curve.