On the diophantine equation xp + 22m = py2

Zhenfu Cao

doi:10.1090/s0002-9939-00-05517-9

On the diophantine equation x^p + 2^2m = p^y2

Zhenfu Cao^*

^*Corresponding author for this work

Harbin Institute of Technology

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

4 Scopus citations

Abstract

Let p be an odd prime. In this paper, using some theorems of Adachi and the author, we prove that if p ≡ 1(mod 4) and p B_(p-1)/2, then the equation x^p + 1 = py², y ≠ 0, and the equation x^p + 2^2m = py², m ∈ ℕ, gcd(x, y) = 1, p | y, have no integral solutions respectively. Here B_(p-1)/2 is (p - 1)/2th Bernoulli number.

Original language	English
Pages (from-to)	1927-1931
Number of pages	5
Journal	Proceedings of the American Mathematical Society
Volume	128
Issue number	7
DOIs	https://doi.org/10.1090/s0002-9939-00-05517-9
State	Published - 2000
Externally published	Yes

Keywords

Adachi's theorem
Bernoulli number
Exponential diophantine equation
Higher degree diophantine equation
Pell's equation

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10.1090/s0002-9939-00-05517-9

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title = "On the diophantine equation xp + 22m = py2",

abstract = "Let p be an odd prime. In this paper, using some theorems of Adachi and the author, we prove that if p ≡ 1(mod 4) and p B(p-1)/2, then the equation xp + 1 = py2, y ≠ 0, and the equation xp + 22m = py2, m ∈ ℕ, gcd(x, y) = 1, p | y, have no integral solutions respectively. Here B(p-1)/2 is (p - 1)/2th Bernoulli number.",

keywords = "Adachi's theorem, Bernoulli number, Exponential diophantine equation, Higher degree diophantine equation, Pell's equation",

author = "Zhenfu Cao",

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journal = "Proceedings of the American Mathematical Society",

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T1 - On the diophantine equation xp + 22m = py2

AU - Cao, Zhenfu

PY - 2000

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N2 - Let p be an odd prime. In this paper, using some theorems of Adachi and the author, we prove that if p ≡ 1(mod 4) and p B(p-1)/2, then the equation xp + 1 = py2, y ≠ 0, and the equation xp + 22m = py2, m ∈ ℕ, gcd(x, y) = 1, p | y, have no integral solutions respectively. Here B(p-1)/2 is (p - 1)/2th Bernoulli number.

AB - Let p be an odd prime. In this paper, using some theorems of Adachi and the author, we prove that if p ≡ 1(mod 4) and p B(p-1)/2, then the equation xp + 1 = py2, y ≠ 0, and the equation xp + 22m = py2, m ∈ ℕ, gcd(x, y) = 1, p | y, have no integral solutions respectively. Here B(p-1)/2 is (p - 1)/2th Bernoulli number.

KW - Adachi's theorem

KW - Bernoulli number

KW - Exponential diophantine equation

KW - Higher degree diophantine equation

KW - Pell's equation

UR - https://www.scopus.com/pages/publications/23044517522

U2 - 10.1090/s0002-9939-00-05517-9

DO - 10.1090/s0002-9939-00-05517-9

M3 - 文章

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SN - 0002-9939

VL - 128

SP - 1927

EP - 1931

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

IS - 7

ER -

On the diophantine equation x^p + 2^2m = p^y2

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