The Pell equation, by Edward Everett Whitford.

64

THE PELL EQUATION

latter equation he deduces "the remarkable theorem
which contains within it the foundations of higher solutions." This theorem is the same as that which the
Hindus had used and resembles the one mentioned above.
If a, b, is a solution of the equation
x2 - Ay2 = SI
and a, 3, is a solution of the equation
x2 - Ay2 = S2,
then
x aa -b Abe, y = a: =~ ba,
satisfies
x2 - Ay2 = SS2.
A more important paper1 of Euler's was published in
1767 in which he solved the equation
x2 - Ay2 = I
by reducing the method of Brouncker to the development
of -A7 into a continued fraction. He showed this development to be symmetric and periodic. He takes up again
the equation
ay2 + by + c = x2,
but goes on to state that "this investigation can be
extended to any quadratic equation between two numbers,
Ay2 + 2Bxy + Cx2 + 2Dy + 2Ex + F = 0,
if one solution is known." Euler does not conceal the
fact that the calculation is complicated and laborious.
Solutions of the Pell equation are here given for values of
A from A = 2 to A = 99 and also for A = 109, 113, 157,
367. He goes on to show that the labor of its solution
can be significantly lightened by the development of IA
into a continued fraction, a process probably the reverse
of that of Archimedes.
1 L. Euler, "De usu novi algorithmi in problemate Pelliano solvendo,"
Novi commentarii Academiae scientiarum imperialis Petropolitanae, 1765,
vol. XI, p. 28, St. Petersburg, 1767.