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A remark on Jeśmanowicz’ conjecture for the non-coprimality case

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Abstract

Let a, b, c be relatively prime positive integers such that a 2 + b 2 = c 2. Jeśmanowicz’ conjecture on Pythagorean numbers states that for any positive integer N, the Diophantine equation (aN)x + (bN)y = (cN)z has no positive solution (x, y, z) other than x = y = z = 2. In this paper, we prove this conjecture for the case that a or b is a power of 2.

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Authors and Affiliations

  1. Department of Mathematics, College of Science and Technology, Nihon University, 1-8-14 Kanda-Surugadai, Chiyoda-ku, Tokyo, 101-8308, Japan

    Takafumi Miyazaki

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  1. Takafumi Miyazaki
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Correspondence to Takafumi Miyazaki.

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Supported by Grant in Aid for JSPS Fellows (Grant No. 25484)

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Miyazaki, T. A remark on Jeśmanowicz’ conjecture for the non-coprimality case. Acta. Math. Sin.-English Ser. 31, 1255–1260 (2015). https://doi.org/10.1007/s10114-015-4491-2

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  • DOI: https://doi.org/10.1007/s10114-015-4491-2

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