The Pell equation, by Edward Everett Whitford.

142

THE PELL EQUATION

lungen zu Geschichte der Mathematik, vol. IX, p. 555,
Leipzig, 1899.
E. CAHEN, "Resolution de l'equation de Pell" in
"Elements de theorie des nombres," p. 221-275, Paris,
1900.
The solutions are based on the theory of quadratic
forms. Theorems are given for both positive and negative
discriminant. From the two positive solutions all the
others are deduced.
A. CUNNINGHAM, "If r and z are two integral solutions
of 2r2 - 2 = 1, then 13 + 33 + 53 + *. + (2r - l)3=r2z2,
Mathematical questions from the Educational Times, vol.
LXXII, p. 45, vol. LXXIII, p. 132, London, 1900.
R. W. D. CHRISTIE, A. CUNNINGHAM, "If p is a prime
of the form 8M + 3, solve X - pY2 = 1 in integers without using the method of continued fractions, generalize
the method for all odd primes, state the cases when the
solution is instanstaneous," Mathematical questions from
the Educational Times, vol. LXXIII, p. 115, London,
1900.
The equation X2 - pY2 = 1 may sometimes be solved
by assuming X, Y, to be functions of other variables
x, y, the transformed equation in x, y, admitting of
easy solution. Many examples are given.
C. DE POLIGNAC, "Solution of question No. 14,713,"
Mathematical questions from the Educational Times,
vol. LXXV, p. 67, 1901.
If ti, ul, is the fundamental solution of the Pell equation t2 - Du2 = 1, and t,, u,, is any other solution,
there is a linear substitution
(Qlx + S1)
xl (Plx + R1)'
such that when we write the nth power
(Qnx + Sn)
axr = (Pn + Rn)