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

Harbin Institute of Technology

Research output: Contribution to journal › Article › peer-review

4 Scopus citations

Abstract

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).

Original language	English
Pages (from-to)	1-4
Number of pages	4
Journal	Proceedings of the Japan Academy Series A: Mathematical Sciences
Volume	77
Issue number	1
DOIs	https://doi.org/10.3792/pjaa.77.1
State	Published - 2001
Externally published	Yes

Keywords

Exponential Diophantine equation

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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).",

keywords = "Exponential Diophantine equation",

author = "Zhenfu Cao and Xiaolei Dong",

year = "2001",

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AU - Dong, Xiaolei

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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 + b^y = c^z

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Keywords

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