A Pell equation is an equation of the form y ² = ax ² + 1, where a is any positive whole number except a square number.

 

The name comes from the English mathematician John Pell (1611–1685); however, he was wrongly credited. In writing about some of the work done on this type of equation, Leonhard Euler gave priority to Pell whereas, in fact, Pell had done no more than copy it in his papers from some of Pierre de Fermat's letters.

 

Fermat had been the first to state that an equation of this form always has an unlimited number of integer solutions. For example, the equation y ² = 92x ² + 1, has the solutions x = 0, y = 1; x = 120, y = 1151; x = 276240, y = 2649601; and so on. Each successive solution is about 2300 times the previous solution. In fact, the solutions are every eighth partial fraction (where x is the numerator and y is the denominator) of the continued fraction for √92.

 

A Pell equation was used in finding the solution to Archimedes' cattle problem.