The Pell equation, by Edward Everett Whitford.

THE PELL EQUATION

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his letter to Frenicle in the same month (February, 1657),
except in that place 109 is the only large number mentioned. These particular numbers, 109, 149, 433, are
evidently chosen for their difficulty, for the fundamental
solutions in these cases are very large numbers.
In the conclusion of his letter to Frenicle, Fermat expressly says: " It appears without saying that my proposition looks only to finding integers which satisfy the
question, for in the case of fractions the lowest type of
arithmetician could see the solution."1 The secretary of
Digby who made the copy for Lord Brouncker may have
regarded the introduction, which expressly set forth the
demand for integers, as unimportant, and so left it uncopied. This would explain why Brouncker, and after
him Wallis, thought that only rational numbers were
demanded.2 They accordingly first handled the problem
by the method used in many places by Diophantus, and
which also resembles one of the rules of Bhaskara. If the
equation
ay2 + 1 = X2
is to be solved for rational values, then the supposition
that
ay2 + 1 = (1 -ry)2
gives the equation
2r
y r2 - a
whence
4r2
= (r2 _ a)2
and if we substitute
r2 = q, r2-a = d,
we have
4q
Y- = d2,
1 Oeuvres de Fermat," vol. II, p. 334.
2 G. Wertheim, "Pierre Fermats Streit mit John Wallis, ein Beitrag
zur Geschichte der Zahlentheorie," Abhandlungen zur Geschichte der
Mathematik, vol. IX, p. 563, Leipzig, 1899.
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