The Diophantine equation x2 + by = cz

Zhenfu Cao; Xiaolei Dong

doi:10.3792/pjaa.77.1

The Diophantine equation x² + b^y = c^z

Zhenfu Cao^*
, Xiaolei Dong

^*此作品的通讯作者

Harbin Institute of Technology

科研成果: 期刊稿件 › 文章 › 同行评审

4 引用（Scopus）

摘要

Let b be an odd prime, m,r ∈ N with 2|m and 2 < r,r > 1, and define the integers U_r, V_r by (m + √-1)^r = V_r + U_r √-1. In this paper, we prove that if a = |V_r|, b = |U_r|, c = m² + 1, and b > 8 · 10⁶, b ≡ 3(mod 4), then the Diophantine equation x² + b^y = c^z has only the positive integer solution (x, y, z) = (a, 2, r).

源语言	英语
页（从-至）	1-4
页数	4
期刊	Proceedings of the Japan Academy Series A: Mathematical Sciences
卷	77
期	1
DOI	https://doi.org/10.3792/pjaa.77.1
出版状态	已出版 - 2001
已对外发布	是

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10.3792/pjaa.77.1

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title = "The Diophantine equation x2 + by = cz",

abstract = "Let b be an odd prime, m,r ∈ N with 2|m and 2 < r,r > 1, and define the integers Ur, Vr by (m + √-1)r = Vr + Ur √-1. In this paper, we prove that if a = |Vr|, b = |Ur|, c = m2 + 1, and b > 8 · 106, b ≡ 3(mod 4), then the Diophantine equation x2 + by = cz has only the positive integer solution (x, y, z) = (a, 2, r).",

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N2 - Let b be an odd prime, m,r ∈ N with 2|m and 2 < r,r > 1, and define the integers Ur, Vr by (m + √-1)r = Vr + Ur √-1. In this paper, we prove that if a = |Vr|, b = |Ur|, c = m2 + 1, and b > 8 · 106, b ≡ 3(mod 4), then the Diophantine equation x2 + by = cz has only the positive integer solution (x, y, z) = (a, 2, r).

AB - Let b be an odd prime, m,r ∈ N with 2|m and 2 < r,r > 1, and define the integers Ur, Vr by (m + √-1)r = Vr + Ur √-1. In this paper, we prove that if a = |Vr|, b = |Ur|, c = m2 + 1, and b > 8 · 106, b ≡ 3(mod 4), then the Diophantine equation x2 + by = cz has only the positive integer solution (x, y, z) = (a, 2, r).

KW - Exponential Diophantine equation

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The Diophantine equation x2 + by = cz

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The Diophantine equation x² + b^y = c^z