The Pell equation, by Edward Everett Whitford.

THE PELL EQUATION

91

or
p = 8n +5,
we can from these values go back to those of t and u in
t2 - pu2 = 1.
Elliptic functions were first used in the solution of the
Pell equations by Kronecker.1 By his method he shows
in a surprising manner how several different algebraic
expressions can be represented approximately by the same
expression. Thus for D = 5, 13 or 37, the expressions
2 + 415, 18+5 413, 882 + 145 -,37,
are all represented approximately by
1 D
8 '
It is interesting to note also the diverse forms which are
obtained by elliptic functions for one and the same solution of the Pell equation. Thus in the solution of
2 - 17y2= - 1,
for the expression 4 + 417, both of these approximate
expressions are obtained:
-2 e- ls 17 1
9 ' /5
Kronecker goes on to say, "The consequences which
spring from the preceding results are very important.
We not only discover a surprising relation between quadratic forms corresponding to opposed determinants, but,
moreover we perceive here the decomposition of equations
into singular moduli, which formed the principal object
1M. Kronecker, "Sur la resolution de l'equation de Pell au moyen des
fonctions elliptiques," translated into French by M. Hotiel, Annales
scientifiques de l'Ecole Normale Superieure, vol. III, p. 303, Paris, 1866,
from the Monatsbericht der Akademie der Wissenschaft, p. 44, Jan. 22,
1863, Berlin.