丢番图方程与实二次域类数的可除性 |
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引用本文: | 曹珍富. 丢番图方程与实二次域类数的可除性[J]. 数学学报, 1994, 37(5): 625-631. DOI: cnki:ISSN:05831431.0.1994-05-007 |
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作者姓名: | 曹珍富 |
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作者单位: | 哈尔滨工业大学数学系 |
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摘 要: | 设d无平方因子,h(d)是二次域的类数。本文证明了:在方程U ̄2-dV ̄2=4,(U,V)=1有整数解时,丢番图方程4x ̄(2n)-dy ̄2=-1,n>2无|y|>1的整数解;如果正整数a,k,n满足,k>1,n>2且而是Pell方程x ̄2-dy ̄2=-1的基本解,则h(d)≡0(modn)。
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关 键 词: | 类数,丢番图方程,实二次域 |
收稿时间: | 1991-10-12 |
修稿时间: | 1993-04-19 |
Diophantine Equations and the Divisibility of the Class Number of the Real Quadra-tic Field |
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Affiliation: | Caozhenfu (Department of Mathematics,Harnin Institute of Technology,Harbin 150006,China) |
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Abstract: | Let d be a positive integer which is square free,h(d)be the class number of thereal quadratic field.In this paper,we prove that(1)if the equation U ̄2-dV ̄2=4 hasinteger solution with(U,V )=1,then the diophantine equation 4x ̄(2n)-dy ̄2=-1,n>2 hasno solution in positive integers,except d = 5,2=y=1;(2)if the positive integers a,k,nsatisfy,n> 2 and is the fundamental solution of Pell′s equationx ̄2-dy ̄2=-1,Then h(d)≡0(mod n). |
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Keywords: | class numbers Diophantine eqnations real quadratic field |
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