The Pell equation, by Edward Everett Whitford.

120

THE PELL EQUATION

of the equations x2 - Ay2 = b are given for A = 2, 3,
5,... 20, b = 1, 2,... 10, giving several solutions for
each A and b; and for powers of b as far as
x2 _ Ay2 = 7r+.1
A. LAISANT, MORET-BLANC, "Trouver un entier tel,
que son carre augmente de 1 soit equal au double d'un
carre," Nouvelles annales de mathematiques, vol. XXVIII, [VIII (2)], p. 336, vol. XXXI, [XI (2)], p. 173,
Paris, 1869, 1872.
N. DE KHANIKOF, "Procede pour resoudre, en nombres
entiers, l'equation indeterminee A + Bt2 = u2," Comptes
rendus de i'Academie, vol. LXIX, p. 185, Paris, 1869.
L. CALZOLARI, "Nota sull'equazione Ax2 =- By2 = u2,
Giornale di matematiche, vol. VIII, p. 28, Naples, 1870.
Such theorems are proved as a necessary and sufficient
condition for the solution in integers of the equation
u2 = Ax2 = By2 is that the trinomial Ab2 - Ba2 =- AB
be a square.
P. SEELING, "fUber die Auflosung der Gleichung
x2 - Ay2 = -= 1 in ganzen Zahlen, wo A positiv und kein
vollstandiges Quadrat sein muss," Archiv der Mathematik und Physik, vol. LII, p. 40, Griefswald, 1870.
At the close of this article there is a table of numbers A
to A = 7,000 for which /IA has an odd period, and therefore for which
x2 - Ay2 = - 1
is solvable. Thus
x2 - 6,997y2 = - 1
has a solution.
A. B. EVANS, A. MARTIN, "To find the smallest integer
y which satisfies the relations 940,751y2 + 1 = o, and
940,751y2 + 38 = o," Mathematical questions from the
Educational Times, vol. XVI, p. 34, London, 1872.
There is no solution in integers to the equation
940,751y2 + 38 = x2.