1 8 N ov 2 00 3 HIGHER DESCENT ON PELL CONICS . II . TWO CENTURIES OF MISSED OPPORTUNITIES
نویسنده
چکیده
It was already observed by Euler [Eul1773] that the method of continued fractions occasionally requires a lot of tedious calculations, and even Fermat knew – as can be seen from the examples he chose to challenge the English mathematicians – a few examples with large solutions. To save work, Euler suggested a completely different method, which allows to compute even very large solutions of certain Pell equations rather easily; its drawback was that the method worked only for a specific class of equations. Although Euler’s tricks were rediscovered on an almost regular basis, nobody really took this approach seriously or generalized it to arbitrary Pell equations. The main goal of [Lem2003a] and this article is to discuss certain results that have been obtained over the last few centuries and which will be put into a bigger perspective in [Lem2003b]. This is opposite to what Dickson aimed at when he wrote his history; in [Dic1920, vol II, preface] he says
منابع مشابه
Higher Descent on Pell Conics. Iii. the First 2-descent
In [Lem2003b] we have sketched the historical development of problems related to Legendre’s equations ar−bs = 1 and the associated Pell equation x−dy = 1 with d = ab. In [Lem2003c] we discussed certain “non-standard” ideas to solve the Pell equation. Now we move from the historical to the modern part: below we will describe the theory of the first 2-descent on Pell conics and explain its connec...
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The theory of Pell’s equation has a long history, as can be seen from the huge amount of references collected in Dickson [Dic1920], from the two books on its history by Konen [Kon1901] and Whitford [Whi1912], or from the books by Weber [Web1939], Walfisz [Wal1952], Faisant [Fai1991], and Barbeau [Bar2003]. For the better part of the last few centuries, the continued fractions method was the und...
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