On the Terai-Jeśmanowicz conjecture

Zhenfu Cao; Xiaolei Dong

doi:10.5486/pmd.2002.2523

On the Terai-Jeśmanowicz conjecture

Zhenfu Cao^*
, Xiaolei Dong

^*Corresponding author for this work

Shanghai Jiao Tong University

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

13 Scopus citations

Abstract

Let a, b, c ∈ ℕ be fixed satisfying a² + b² = c^r with gcd(a, b) = 1 and r odd ≥ 3. In this paper, we prove that (A) if b ≡ 3 (mod 4), 2∥a and b ≥ 25.1a, then the Diophantine equation (1) a^x + b^y = c^z has only the positive integer solution (x, y, z) = (2, 2, r); (B) if a = |V_r|, b = |U_r|, c = m² + 1, where the integers U_r, V_r satisfy (m + √-1)^r = V_r + U_r√-1, and b ≡ 3 (mod 4), 2∥a and b is a prime, then equation (1) has only the positive integer solution (x, y, z) = (2, 2, r).

Original language	English
Pages (from-to)	253-265
Number of pages	13
Journal	Publicationes Mathematicae Debrecen
Volume	61
Issue number	3-4
DOIs	https://doi.org/10.5486/pmd.2002.2523
State	Published - 2002
Externally published	Yes

Keywords

Exponential diophantine equation

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title = "On the Terai-Je{\'s}manowicz conjecture",

abstract = "Let a, b, c ∈ ℕ be fixed satisfying a2 + b2 = cr with gcd(a, b) = 1 and r odd ≥ 3. In this paper, we prove that (A) if b ≡ 3 (mod 4), 2∥a and b ≥ 25.1a, then the Diophantine equation (1) ax + by = cz has only the positive integer solution (x, y, z) = (2, 2, r); (B) if a = |Vr|, b = |Ur|, c = m2 + 1, where the integers Ur, Vr satisfy (m + √-1)r = Vr + Ur√-1, and b ≡ 3 (mod 4), 2∥a and b is a prime, then equation (1) has only the positive integer solution (x, y, z) = (2, 2, r).",

keywords = "Exponential diophantine equation",

author = "Zhenfu Cao and Xiaolei Dong",

year = "2002",

doi = "10.5486/pmd.2002.2523",

language = "英语",

volume = "61",

pages = "253--265",

journal = "Publicationes Mathematicae Debrecen",

issn = "0033-3883",

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number = "3-4",

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TY - JOUR

T1 - On the Terai-Jeśmanowicz conjecture

AU - Cao, Zhenfu

AU - Dong, Xiaolei

PY - 2002

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N2 - Let a, b, c ∈ ℕ be fixed satisfying a2 + b2 = cr with gcd(a, b) = 1 and r odd ≥ 3. In this paper, we prove that (A) if b ≡ 3 (mod 4), 2∥a and b ≥ 25.1a, then the Diophantine equation (1) ax + by = cz has only the positive integer solution (x, y, z) = (2, 2, r); (B) if a = |Vr|, b = |Ur|, c = m2 + 1, where the integers Ur, Vr satisfy (m + √-1)r = Vr + Ur√-1, and b ≡ 3 (mod 4), 2∥a and b is a prime, then equation (1) has only the positive integer solution (x, y, z) = (2, 2, r).

AB - Let a, b, c ∈ ℕ be fixed satisfying a2 + b2 = cr with gcd(a, b) = 1 and r odd ≥ 3. In this paper, we prove that (A) if b ≡ 3 (mod 4), 2∥a and b ≥ 25.1a, then the Diophantine equation (1) ax + by = cz has only the positive integer solution (x, y, z) = (2, 2, r); (B) if a = |Vr|, b = |Ur|, c = m2 + 1, where the integers Ur, Vr satisfy (m + √-1)r = Vr + Ur√-1, and b ≡ 3 (mod 4), 2∥a and b is a prime, then equation (1) has only the positive integer solution (x, y, z) = (2, 2, r).

KW - Exponential diophantine equation

UR - https://www.scopus.com/pages/publications/0036457744

U2 - 10.5486/pmd.2002.2523

DO - 10.5486/pmd.2002.2523

M3 - 文章

AN - SCOPUS:0036457744

SN - 0033-3883

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SP - 253

EP - 265

JO - Publicationes Mathematicae Debrecen

JF - Publicationes Mathematicae Debrecen

IS - 3-4

ER -

On the Terai-Jeśmanowicz conjecture

Abstract

Keywords

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