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<p>In this note, we prove that there do not exist three consecutive powerful numbers, one of which is of the form <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(x^{n}\pm 1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>x</mi> <mi>n</mi> </msup> <mo>±</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and whose <i>cubic square-free part</i> (i.e., the unique square-free integer <i>b</i> such that the number can be written as <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(a^{2}b^{3}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>a</mi> <mn>2</mn> </msup> <msup> <mi>b</mi> <mn>3</mn> </msup> </mrow> </math></EquationSource> </InlineEquation>) is either 1 or the product of distinct primes.</p>

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A remark on three consecutive powerful numbers

  • 期刊论文

  • 发表日期：

    2026年5月27日

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  • Tran Duc Son

摘要

In this note, we prove that there do not exist three consecutive powerful numbers, one of which is of the form \(x^{n}\pm 1\) x n ± 1 and whose cubic square-free part (i.e., the unique square-free integer b such that the number can be written as \(a^{2}b^{3}\) a 2 b 3 ) is either 1 or the product of distinct primes.