The Pell equation, by Edward Everett Whitford.

THE PELL EQUATION

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they change to
s1 =1, S2 = s1 + 1=2, 83 = S2 + 62 = 5,..
51 = 1, 52 = 2si + 61 = 3, ~3 = 2S2 + 82 = 7, *.
Then no longer, as in the first double series, does the
square of each lower member equal double the square of
the upper, but the difference is constantly 1 and alternately + and -. Each 5r, Sr, is a solution of one of the
Pell equations
2 - 2s2 = _ 1.
Theon1 of Smyrna, who lived about 130 A.D., set forth
this development earlier than Proclus. He called these
numbers side- and diagonal numbers, and both he and
Proclus carried the computation as far as
2.122 + 1 = 172.
Theon evidently understood the generality of the double
series for he said, "In this manner the diagonals and sides
will always be expressible," perhaps'omitting the general
proof2 because it was so evident. Expressed in algebraic
symbols, if
X2 - 2y2 = _ 1,
suppose x = p, y = q and put
xl = p + 2q, yl = p + q;
then
X12 2y12 = p2 + 4pq + 4q 2- 2p2 - 4pq - 2q2
= - p2 + 2q2 = - (p2 - 2q2).
But evidently
x"=y=l
is a first solution of x2 - 2y2 - 1. Therefore
X12-2y~2 = =F 1.
Theonis Smyrnaei, "Platonici, eorum, quae in mathematicis ad
Platonis lectionem utilia sunt, expositio," p. 67, Lutetiae Parisiorum, 1544.
"Oeuvres de Theon de Smyrne," J. Dupuis, editor, p. 71, Paris, 1892.
2G. H. F. Nesselmann, "Die Algebra der Griechen," p. 229, Berlin,
1842. Compare M. Cantor, vol. I, p. 408, 2d ed.