一类Brahmagupta-Fermat-Pell方程x^2-dy^2=±1(英文)

The Brahmagupta-Fermat-Pell Equation x^2-dy^2=±1

This paper corresponds to the written versions of many lectures at several locations including the most recent one at Weinan Teachers University on June 8,2011.I would like to thank Professor Hailong Li for inviting me to publish this in the journal of his university.I wish also to express my deep gratitude to my friend Shigeru Kanemitsu,thanks to whom I could visit Weinan Teachers University,and who also came up with a written version of these notes. The topic is centered around the equation x2-dy2=±1,which is important because it produces the(infinitely many) units of real quadratic fields.This equation,where the unknowns x and y are positive integers while d is a fixed positive integer which is not a square,has been mistakenly called with the name of Pell by Euler.It was investigated by Indian mathematicians since Brahmagupta(628) who solved the case d=92,then by Bhaskara II(1150) for d=61 and Narayana(during the 14-th Century) for d=103.The smallest solution of x2-dy2=1 for these values of d are respectively 1 1512-92·1202=1, 1 766 319 0492-61·226 153 9802=1 and 227 5282-103·22 4192=1, and hence they could not have been found by a brute force search! After a short introduction to this long story of Pell's equation,we explain its connection with Diophantine approximation and continued fractions(which have close connection with the structure of real quadratic fields),and we conclude by saying a few words on more recent developments of the subject in terms of varieties.Finally we mention applications of continued fraction expansion to electrical circuits.

This paper corresponds to the written versions of many lectures at several locations including the most recent one at Weinan Teachers University on June 8, 2011. I would like to thank Professor Hailong Li for inviting me to publish this in the journal of his university. I wish also to express my deep gratitude to my friend Shigeru Kanemitsu, thanks to whom I could visit Weinan Teachers University, and who also came up with a written version of these notes. The topic is centered around the equation x2 - dy2 = ＋ 1 , which is important because it produces the （infinitely many） units of real quadratic fields. This equation, where the unknowns x and y are positive integers while d is a fixed positive integer which is not a square, has been mistakenly called with the name of Pell by Euler. It was investigated by Indian mathematicians since Brahmagupta （628） who solved the case d = 92, then by Bhaskara Ⅱ （1150） for d = 61 and Narayana （during the 14-th Century） for d = 103. The smallest solution of x2 - dy2 = 1 for these values of d are respectively 1 1512-92· 1202 = 1, 1 766 319 0492-61 -226 153 9802 = 1 and 227 5282 - 103 · 22 4192 = 1, and hence they could not have been found by a brute force search! After a short introduction to this long story of Pell＇s equation, we explain its connection with Diophantine approximation and continued fractions （ which have close connection with the structure of real quadratic fields）, and we conclude by saying a few words on more recent developments of the subject in terms of varieties. Finally we mention applications of continued fraction expansion to electrical circuits.