| 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. |
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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： | 1993-4-19 |
Diophantine Equations and the Divisibility of the Class Number of the Real Quadra-tic Field |
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| Institution: | 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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