On Terai's conjecture

Zhenfu Cao; Xiaolei Dong

doi:10.3792/pjaa.74.127

On Terai's conjecture

Zhenfu Cao^*
, Xiaolei Dong

^*Corresponding author for this work

Harbin Institute of Technology

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

12 Scopus citations

Abstract

Terai presented the following conjecture: If a² + b² = c² with a > 0, b > 0, c > 0, gcd (a, b, c) = 1 and a even, then the diophantine equation x² + b^m = cⁿ has the only positive integral solution (x, m, n) = (a, 2, 2). In this paper we prove that if (i) b is a prime power, c ≡ 5 (mod 8), or (ii) c ≡ 5 (mod 8) is a prime power, then Terai's conjecture holds.

Original language	English
Pages (from-to)	127-129
Number of pages	3
Journal	Proceedings of the Japan Academy Series A: Mathematical Sciences
Volume	74
Issue number	8
DOIs	https://doi.org/10.3792/pjaa.74.127
State	Published - 1998
Externally published	Yes

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title = "On Terai's conjecture",

abstract = "Terai presented the following conjecture: If a2 + b2 = c2 with a > 0, b > 0, c > 0, gcd (a, b, c) = 1 and a even, then the diophantine equation x2 + bm = cn has the only positive integral solution (x, m, n) = (a, 2, 2). In this paper we prove that if (i) b is a prime power, c ≡ 5 (mod 8), or (ii) c ≡ 5 (mod 8) is a prime power, then Terai's conjecture holds.",

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AU - Dong, Xiaolei

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AB - Terai presented the following conjecture: If a2 + b2 = c2 with a > 0, b > 0, c > 0, gcd (a, b, c) = 1 and a even, then the diophantine equation x2 + bm = cn has the only positive integral solution (x, m, n) = (a, 2, 2). In this paper we prove that if (i) b is a prime power, c ≡ 5 (mod 8), or (ii) c ≡ 5 (mod 8) is a prime power, then Terai's conjecture holds.

UR - https://www.scopus.com/pages/publications/22444451769

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M3 - 文章

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JO - Proceedings of the Japan Academy Series A: Mathematical Sciences

JF - Proceedings of the Japan Academy Series A: Mathematical Sciences

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ER -

On Terai's conjecture

Abstract

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