The Pell equation, by Edward Everett Whitford.

THE PELL EQUATION

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have known the solution of the equation x2 - Ay2 = 1.
At least he proposed this problem as a challenge to the
English mathematicians. It is Lord Brouncker's solution which we find in the works of Wallis and which is
contained almost word for word in the second part of
Euler's algebra. But on the one hand Fermat has made
public nothing of his own solution, and on the other
hand the process of the English mathematicians, although
it is very ingenious, does not show in a definite manner
that the problem is always solvable. It remained therefore to prove that the equation x2 - Ay2 = 1 can always
be solved in integers. Lagrange1 has done this in a
sagacious as well as rigid manner.... This proof, as well
as the one added by him, must be considered the most
important step which has been made up to the present
time in the indeterminate analysis."
Gauss2 expressed himself in the following manner:
"This famous problem (to solve in integers all indeterminate equations of the second degree) Lagrange3 has
completely solved. There is also an inferior complete
solution by Lagrange in the supplement to Euler's
algebra. The treatise of Lagrange grasps the problem
in its entire generality and in this connection leaves
nothing to be desired."
Lagrange4 himself sums up his opinion in the following words: "It is in truth, very surprising that M. de
Fermat who has been for so long a time and with such
success occupied with the theory of integral numbers
has not sought to solve generally the indeterminate
problem of the second and higher degrees, as M. Bachet
1 J. L. Lagrange, "Solution d'un probleme d'arithmetique," Miscellanea Taurinensis, vol. IV, p. 41, Turin, 1766. "Oeuvres de Lagrange,"
vol. I, p. 671, Paris, 1867.
2 C. F. Gauss, "Disquisitiones arithmeticae," ~ 222, p. 309, Leipzig, 1801.
3 J. L. Lagrange, "Histoire de l'Academie de Berlin," p. 165, 1767, p.
181, 1768.
4 J. L. Lagrange, Memoires de l'Academie de Berlin, vol. XXIV, p. 236,
1768.