The Pell equation, by Edward Everett Whitford.

48

THE PELL EQUATION

"Now arithmetic has a special domain of its own, the
theory of numbers. This was touched upon but only to
a slight degree by Euclid in his Elements, and by those
who followed him it has not been sufficiently extended,
unless perchance it lies hid in those books of Diophantus
which the ravages of time have destroyed. Arithmeticians
['AptOLTr)Lciwv rakSes] have now to develop or restore it.
"To these, that I may lead the way, I propose this
theorem to be proved or problem to be solved. If they
succeed in discovering the proof or solution, they will
acknowledge that questions of this kind are not inferior
to the more celebrated ones from geometry either for
depth or difficulty or method of proof:
"Given any number which is not a square, there also
exists an infinite number of squares such that when
multiplied into the given number and unity is added to
the product, the result is a square."
The problem thus set forth by Fermat is one of the
most important steps in the history of the Pell equation.
A freer translation of the Latin would read: For every
given number which is not a square there exists infinitely
many square numbers such that the product of each by
the given number, with the addition of 1, is a square.
Fermat illustrates his problem by a number of examples,
one of which is as follows:
"Given 3, a non-square number; this number multiplied into the square number 1 and 1 being added produces
4, which is a square. Moreover, the same 3 multiplied
into the square 16 with 1 added makes 49, which is a
square. And instead of 1 and 16, an infinite number of
squares may be found showing the same property; I demand, however, a general rule, any number being given
which is not a square. It is sought, for example, to find
a square which when multiplied into 149, 109, 433, etc.,
becomes a square when unity is added."
The above is very much like the problems set forth in