The Pell equation, by Edward Everett Whitford.

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THE PELL EQUATION

R. W. D. CHRISTIE, E. B. ESCOTT, "Prove that the socalled Pellian equation may be reversed by the dual principle as follows: Since
(4a3 + 6a2 + 6a + 2)2 - (4a2 + 4a + 5)(2a2 + 2a + 1)2=- 1
for all values of the letter, then
(2a3 + 6a2 + 6a + 4)2 - (5a2 + 4a + 4)(a2 + 2a + 2)2= -a6.
Give a few other examples and show how to introduce the
prime roots," Mathematical questions from the Educational Times, vol. VI (2), p. 119, London, 1904.
A general solution is
{k(4n2a2:= 4na + 4n2 + 1) + (2na =F a + 2n)}2 + 1
= {(2na =F 1)2 + (2n)2} {(2kn + 1)2 + (2kna + k -+ a)2}2,
where all the letters are arbitrary.
P. F. TEILHET, A. BOUTIN, "The form 632 + 1 is not a
square if 13 is a root of 72 - 332 = 1," L'Intermediare des
mathematiciens, vol. XI, p. 68, 182, Paris, 1904.
E. B. ESCOTT, "Solutions de l'equation
2 - Ay2 =- a,"
L'Intermediare des mathematiciens, vol. XI, p. 156,
Paris, 1904.
A condition necessary that the equation
(ax - by)2 - Dy2 = a
may be solvable in integers is
( a' a
where a', a", a"',... are odd prime factors of a; but this
condition is not sufficient.
G. FRATTINI, "Applicazione di un concetto nuovo all'
analisi indeterminata aritmetica e algebrica di 2~ grado