A new conjecture concerning the Diophantine equation x2 + by = cz

Zhenfu Cao; Xiaolei Dong; Zhong Li

doi:10.3792/pjaa.78.199

A new conjecture concerning the Diophantine equation x² + b^y = c^z

Zhenfu Cao, Xiaolei Dong, Zhong Li

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

6 Scopus citations

Abstract

In this paper, using a recent result of Bilu, Hanrot and Voutier on primitive divisors, we prove that if a = |V_r|, b = |U_r|, c = m² + 1, and b ≡ 3 (mod 4) is a prime power, then the Diophantine equation x² + b^y = c^z has only the positive integer solution (x,y, z) = (a, 2, r), where r > 1 is an odd integer, m ∈ N with 2 | m and the integers U_r, V_r satisfy (m + √-1)^r = V_r + U_r√-1.

Original language	English
Pages (from-to)	199-202
Number of pages	4
Journal	Proceedings of the Japan Academy Series A: Mathematical Sciences
Volume	78
Issue number	10
DOIs	https://doi.org/10.3792/pjaa.78.199
State	Published - Dec 2002
Externally published	Yes

Keywords

Exponential Diophantine equation
Gauss integer
Lucas sequence
Primitive divisor

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abstract = "In this paper, using a recent result of Bilu, Hanrot and Voutier on primitive divisors, we prove that if a = |Vr|, b = |Ur|, c = m2 + 1, and b ≡ 3 (mod 4) is a prime power, then the Diophantine equation x2 + by = cz has only the positive integer solution (x,y, z) = (a, 2, r), where r > 1 is an odd integer, m ∈ N with 2 | m and the integers Ur, Vr satisfy (m + √-1)r = Vr + Ur√-1.",

keywords = "Exponential Diophantine equation, Gauss integer, Lucas sequence, Primitive divisor",

author = "Zhenfu Cao and Xiaolei Dong and Zhong Li",

year = "2002",

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AU - Cao, Zhenfu

AU - Dong, Xiaolei

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N2 - In this paper, using a recent result of Bilu, Hanrot and Voutier on primitive divisors, we prove that if a = |Vr|, b = |Ur|, c = m2 + 1, and b ≡ 3 (mod 4) is a prime power, then the Diophantine equation x2 + by = cz has only the positive integer solution (x,y, z) = (a, 2, r), where r > 1 is an odd integer, m ∈ N with 2 | m and the integers Ur, Vr satisfy (m + √-1)r = Vr + Ur√-1.

AB - In this paper, using a recent result of Bilu, Hanrot and Voutier on primitive divisors, we prove that if a = |Vr|, b = |Ur|, c = m2 + 1, and b ≡ 3 (mod 4) is a prime power, then the Diophantine equation x2 + by = cz has only the positive integer solution (x,y, z) = (a, 2, r), where r > 1 is an odd integer, m ∈ N with 2 | m and the integers Ur, Vr satisfy (m + √-1)r = Vr + Ur√-1.

KW - Exponential Diophantine equation

KW - Gauss integer

KW - Lucas sequence

KW - Primitive divisor

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A new conjecture concerning the Diophantine equation x² + b^y = c^z

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Keywords

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