如何求解pell方程x^2-d*y^2=M

nyy

Solving the diophantine equation x2 – Dy2 = N, D > 0 and not a perfect square, N ≠ 0
This first finds all primitive fundamental solutions, then all fundamental solutions for the diophantine equation x2 – Dy2 = N, D > 0, D not a perfect square, N ≠ 0. We have to find all primitive fundamental solutions of the equations x2 – Dy2 = N/f2 for all divisors f of N such that f2 divides N; these belong to classes P, where P2  D (mod |N| ).
E = 1 is verbose and prints partial and complete quotients, as well as convergents, up to the end of a period (or double period, in the case of odd period).
Our algorithm (LMM) goes back to Lagrange 1770 and should be better known, as it generalises the well-known continued fraction algorithm for solving Pell's equation. (See a slide-talk (pdf) by Keith Matthews.)

Another approach to the algorithm, using ideals, was discovered by Richard Mollin - see Expositiones Math. 19 (2001) 55-73.

The standard method is due to Gauss - see G.B. Mathews Number Theory, page 97 or John Robertson, Computing in quadratic orders, page 14.

The book L'équation diophantienne du second degré, Alain Faisant, Hermann 1991, has an algorithm for getting all solutions, which also uses continued fractions.

Enter D (< 1020):
61

Enter N (≠ 0):
97

Enter E (0 or 1):
1

GO RESET
Last modified 7th March 2015

http://www.numbertheory.org/php/patz.html

nyy

上面的链接里面有一个pdf，详细的讲了如何求解

nyy

kastin 发表于 2019-3-19 18:18
见https://bbs.emath.ac.cn/thread-15533-1-1.html
以及http://mathworld.wolfram.com/PellEquation.html最 ...

第1个链接怎么没了？

nyy

pell方程，佩尔方程，x^2-31y^2=8837，素数
试试这个

nyy

我看懂了华罗庚的解答，
他是通过不断降低N，形成一个多米诺骨牌，
然后N的绝对值＜根号d，这时候方程可解，
再把解倒退回去，就得到原方程的解！

nyy

华罗庚的求解办法不好，
需要多次求解剩余方程，
然后求出来的解未必是整数解。

nyy

https://bbs.emath.ac.cn/forum.ph ... =2836&pid=88108

愿力大于一切！
只要想知道如何求解，就有一天能解决！
我现在解决了这个问题