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<p>Starting with Ramanujan’s famous taxicab problem, we can study the solvability of the equations <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(p^n+q^n=r^n+s^n\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>p</mi> <mi>n</mi> </msup> <mo>+</mo> <msup> <mi>q</mi> <mi>n</mi> </msup> <mo>=</mo> <msup> <mi>r</mi> <mi>n</mi> </msup> <mo>+</mo> <msup> <mi>s</mi> <mi>n</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> and, more generally, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(p_1^{k_1}+\dots +p_m^{k_m}=0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>p</mi> <mn>1</mn> <msub> <mi>k</mi> <mn>1</mn> </msub> </msubsup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msubsup> <mi>p</mi> <mi>m</mi> <msub> <mi>k</mi> <mi>m</mi> </msub> </msubsup> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> among polynomials.</p>

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Ramanujan, the taxicab problem for polynomials, and the abc-conjecture

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  • 发表日期：

    2026年1月14日

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  • Katalin Gyarmati

摘要

Starting with Ramanujan’s famous taxicab problem, we can study the solvability of the equations \(p^n+q^n=r^n+s^n\) p n + q n = r n + s n and, more generally, \(p_1^{k_1}+\dots +p_m^{k_m}=0\) p 1 k 1 + ⋯ + p m k m = 0 among polynomials.