Pell’s Equation
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 ${x^2} - d{y^2} = k$, 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 ${x^2} - d{y^2} = 1$ 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 ìPell’s equation” and the name stuck.
