The Pell equation, by Edward Everett Whitford.

THE PELL EQUATION

93

If in the notation of hyperbolic functions, we put
= cosh-' () sinh- ( I),
we shall have
Tn = - cosh no, Un ID = a sinh np.
The convenience of this is that the known formulas of
hyperbolic functions may be used to express the relations
between the different values of Tn and U,, of the equation
T2 - DUn2 = -2
Thus from the formulas,
cosh 2sp = 2 cosh2 p - 1 sinh 2(p = 2 sinh -cosh cp,
we deduce
oT2n = 2Tn2 - 72, oU2n = 2TnUn,
and so on in other cases. It may be specially observed
that
2T 2T
Tn+1 = T n -T 1, Unl1 = - Un - Un-1.
These formulas are very convenient for calculating the
successive values of T and U.1
Several classes of fundamental solutions of x2 -Ay2 = 1
have been noted in which the corresponding values of
A, x, y, occur in arithmetical progressions. Thus, for
example, one class2 would be
A = 5, 10, 17, *,
x = 9, 19, 33,.,
y 4, 6, 8,.,
in which
A = m2 + 4m + 5,
x = 2m2 + 8m + 9 (m = 0, 1, 2,* *),
1 G. B. Mathews, "Theory of Numbers," Part 1, p. 93, Cambridge, 1892.
2 G. Speckman, "Fundamental auflisungen der Pell'schen Gleichung,"
Archiv der Mathematik und Physik, vol. XIII (2), p. 327, Leipzig, 1895.