Pell’s Equation – Phương trình Pell

Pell’s equation seems to be an ideal topic to lead college students, as well as some talented and motivated high school students, to a better appreciation of the power of mathematical technique. The history of this equation is long and circuituous. It involved a number of different approaches before a definitive theory was found. Numbers have fascinated people in various parts of the world over many centuries. Many puzzles involving numbers lead naturally to a quadratic Diophantine equation (an algebraic equation of degree 2 with integer coefficients for which solutions in integers are sought), particularly ones of the form [Hình: latex.php?latex=x%5E2-dy%5E2%3Dk&;amp;s=0], where d and k are integer parameters with d nonsquare and positive. A few of these appear in Chapter 2. For about a thousand years, mathematicians had various ad hoc methods of solving such equations, and it slowly became clear that the equation [Hình: latex.php?latex=x%5E2-dy%5E2%3D1&;amp;s=0] should always have positive integer solutions other than (x, y) = (1, 0). There were some partial patterns and some quite effective methods of finding solutions, but a complete theory did not emerge until the end of the eighteenth century. It is unfortunate that the equation is named after a seventeenth-century English mathematician, John Pell, who, as far as anyone canb tell, had hardly anything to do with it. By his time, a great deal of spadework had been done by manyWestern European mathematicians. However, Leonhard Euler, the foremost European mathematician of the eighteenth century, who did pay a lot of attention to

the equation, referred to it as “Pell’s equation” and the name stuck.


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