The Pell equation, by Edward Everett Whitford.

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THE PELL EQUATION

an Englishman devised a unique method of solving
problems of this kind as shown in the works of Wallis... according to which at the beginning of my work I
have that 1 + aX2 makes a square. But this method was
only available for numerical numbers, and not for formulas
having arbitrary coefficients."
In the same year,1 Euler set forth a method for reducing
the solution of
ay2 + by + c = x2
to that of the Pell equation
x2 - Ay2 = 1.
He gave a list of solutions up to A = 68. He also proved
that the successive solutions of
ay2 + by + c = 2,
when one is known, require that one solution of
av2 + 1 = u2
must also be known. Given one value of y, say n, which
makes ay2 + by + c a perfect square, and one value of v,
say q, which makes av2 + 1 a perfect square, that is,
when
(1) an2 + bn + c = m2
and
aq2 + 1 = p2,
Euler finds any number of solutions of
ay2 + by + c = x2
and the law for forming them. He then takes the particular case
ay2 + by + d2 = x2
where (since
y = 0, x= d
L. Euler, "De solutione...," loc. cit. A. Aubry, "L'oeuvre arithm6tique d'Euler," L'enseignement mathematique, vol. XI, p. 329, Paris,
1909. T. L. Heath, "Diophantus," 2d ed., p. 288, Cambridge, 1910.