The Pell equation, by Edward Everett Whitford.

THE PELL EQUATION

53

where v1 is an integer less than U. Substituting this
value of T in (1), we have
(2) Q(U, v) = 1,
likewise in integers, and where Q signifies a homogeneous
quadratic form. From (2) we have
p1Il < U < (P1 + 1)vi,
U = pliV + V2; V2 < Vi; Qi(vl, v2) = 1,
and so on continually. Since by hypothesis, there is an
integral solution of
x2 - Dy2 = 1,
and since the numbers,
T, U V, vl, v2, *
form a descending series
T > U > vI > v2 **-,
then sometime we must reach the equation
Qn(vn, Vn+l) = 1,
which gives
Vn = PnVn+l;
and, by reason of the hypothesis of the solvability of
X2 - Dy2 = 1,
this also gives an integral value of v,,+,. Through substitution we obtain the values of T and U in a manner
similar to the algorism for obtaining the greatest common
divisor of two integers. For example, consider the
equation
T2 - 13U2 = 1.
We see that
P = 3,
whence
3U< T<4U; T= 3U+vl;
Q(U, vl) = - 4U2 + 6Uvl + v12 = 1.