The Pell equation, by Edward Everett Whitford.

46

THE PELL EQUATION

give solutions to the equation
x2 - 66y2 = 1,
the last one being 81-2-, from which
8,4492 - 66.1,0402 = 1.
Buteo also makes use of the method of Chuquet with the
two series of fractions, the one ascending, the other
descending, which have just been described. For /13
he gives the approximation 5, and this corresponds to
the minimum solution of the equation
x2 - 13y2 = - 1.
Fermat1 was the first to assert that the equation
x2 - Ay2 = 1
where A is any non-square integer, always has an unlimited number of solutions in integers. This equation
may have been suggested to him by the study of some of
the double equations of Diophantus;2 for Fermat says
in a note on Diophantus, IV, 39, "Suppose, if you will,
that the double equation to be solved is
2m + 5 = square,
6m + 3 = square.
The first square must be made equal to 16 and the second
to 36; and others will be found ad infinitum satisfying
the equation. Nor is it difficult to propound a general
rule for the solution of this kind of equation." By elimination the two equations just mentioned lead to
2 - 3y2 = - 12.
"Oeuvres de Fermat publiees par les soins de MM. Paul Tannery et
Charles Henry," vol. II, p. 334, Paris, 1894.
2 For a different view of Fermat's methods of solving the double equations see Paul von Schaewen, "Jacobi de Billy, Doctrinae analyticae
inventum novum, Fermats Briefen an Billy entnommen," p. 61, Berlin,
1910.