The Pell equation, by Edward Everett Whitford.

THE PELL EQUATION

59

and a square is a factor of double the square root of the
square. For example, if the given number is 33 or 39
of which the difference from 36 is 3, and this 3 is a factor
of 12, which is double the square root of 36. For 39,
say,
39x2 + 1 = 36x2 + 12x + 1.
Therefore
3x = 12, x = 4, x2 = 16,
and
39x2 + 1 = 625, etc.
His manner of finding an infinity of solutions from a
given one is perfectly general.
The honor of having first recognized the deep importance of the Pell equation for the general solution of the
indeterminate equation of the second degree belongs to
Euler. He left several memoirs relating to this subject.'
In a letter to Goldbach, August 10, 1732, Euler2 mentions
the equation
x2 - Ay2 = 1
as necessary in order to make ay2 + by + c a square,
and goes on to say: "Problems of this kind have been
discussed between Wallis and Fermat.... The most
difficult example was to find numbers which put in place
of y would make 109y2 + 1 a perfect square. Dr. Pell3
1L. Euler, "De solutione problematum Diophantaeorum per numeros
integros," Commentarii Academiae scientiarum imperialis Petropolitanae,
1732, vol. VI, p. 175, St. Petersburg, 1738. "De resolutione formularum
quadraticarum indeterminatarum per numeros integros," Novi commentarii Academiae scientiarum imperialis Petropolitanae, vol. IX, p. 3,
St. Petersburg, 1764. "De usu novi algorithmi in problemate Pelliano
solvendo," Novi commentarii Academiae scientiarum imperialis Petropolitanae, vol. XI, p. 28, St. Petersburg, 1765.
2 P. H. Fuss, "Correspondance mathematique et physique de quelques
celebres geometres du XVIIIieme siecle," p. 37, St. Petersburg, 1843.
3 Note here probably the first reference of the equation to Pell. For a
suggestion that this mistake of Euler was due to cursory reading, note
also that Euler makes a mistake as to the most difficult cases solved by
Wallis and Brouncker, these really being the equations
433y2 + 1 = x2
and
313y2 + 1 = x2