The Pell equation, by Edward Everett Whitford.

THE PELL EQUATION

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and the successive values of P/Q are closer and closer
approximations to 6. It will be noted that this method
of finding approximations to!a is the same as that which,
according to the hypothesis of Zeuthen and Tannery,
Archimedes used for ~/3.
In 1759 Euler1 stated the theorem that the product of
two expressions of the form x2 + Ay2 is of the same form;
but Goldbach seems to have first remarked this important
theorem in' a letter to Euler in 1753. If we have the
identity
(a2 + Ab) (a2 + Ab2) = (aa =+- Abf3)2 + A(aO3 ba)2,
it follows that if a, b, is one solution of the Pell equation
x2 - Ay2 = 1
and a, 3, is a solution of the equation
x2 - Ay2 = B,
then we will have
x = aa Ab/3, y = a3: ba,
as another solution of the latter equation. But we have
seen that the Hindus knew this remarkable formula at
least a thousand years earlier, so that it was merely a
case of independent rediscovery.
Euler, in letters2 to Goldbach in 1753 and in 1755, mentions certain improvements on the Pellian methods; and
in his next paper3 on this subject, he again speaks of
finding all the solutions of
ay2 + by + c = X2,
or of
ay2 + b = x2,
when one solution is known. In the discussion of the
1L. Euler, in Novi commentarii Academiae scientiarum imperialis
Petropolitanae, 1759, vol. VII, p. 3, St. Petersburg, 1761.
2 P. H. Fuss, op. cit., p. 614, p. 629.
3 L. Euler, "De resolutione...," loc. cit.