The Pell equation, by Edward Everett Whitford.

THE PELL EQUATION

97

= 629, read x = 7,850; N = 667, read y = 4,147,668;
N = 271, x should end with* * 983,600; N = 749, x should
end with... 84,895; N = 751, x should end with... 424,418,960; N = 823, insert1 47 after 235,170 in x; N = 809,
x should begin with 43,385* -..2
The British Association report3 for 1893 contains a
table which continues the solution of one or the other of
the equations y2 - ax2 = - 1 and y2 - ax2 = 1 from
a = 1,001 up to a = 1,500, for all non-square values of a.
The solution of the former equation is given when possible,
of the latter in other cases, and there is the inconvenience
of having the two solutions given at the same time. The
only method of showing that the solution is for the
equation y2 -x2 = - 1 is the placing of an asterisk (*)
after the- argumnent, and even this is omitted under a =
1,361.
Allan Ju.lningham4 has published tables of the fundamental solutions of r2 - Dv2 = 1 for all non-square values
of D <4100, and for the same range, when the solution
exists, for the equation r2 - Dv2 = - 1. He also gives
tables of multiple solutions, To,vo; z1, v1; T2, V2;... of
both the Pell equations, for all non-square values of D
from 2 to 20. Barlow5 gave a table of solutions of
p2 - Nq2 = 1 for every non-square value of N from 2 to
102. For other isolated solutions see the bibliography.
1 H. Richaud, Journal de mathematiques 6elmentaires, p. 183, vol. XI,
Paris, 1887.
2 Richaud, loc. cit. For N = 629 see E. Catalan, Atti dell' Accademia
pontificia de' Nuovi Lincei, vol. XX, p. 3, Rome, 1867; and C. A. Roberts
also makes a correction, Mathematical magazine, vol. II, p. 105, Washington, 1892.
3 C. E. Bickmore, "Report of a committee appointed for the purpose
of carrying on the tables connected with the Pellian equation from the
point where the work was left by Degen in 1817" (text by Professor
Cayley), British Association Report, vol. LXIII, p. 73, Nottingham, 1893,
London, 1894. A. Cayley, "Collected mathematical papers," vol. XIII,
p. 430, Cambridge, 1897.
4 A. Cunningham, " Quadratic partitions," London, 1904.
5 P. Barlow, "An elementary investigation of the theory of numbers,"
p. 507, London, 1811.