The Pell equation, by Edward Everett Whitford.

THE PELL EQUATION

57.

men and has shown that England's champions in wisdom
are just as strong as those in war."
It was months before Fermat was informed of the
details of the solution. This was partly on account
of the first misunderstanding of the demand for integers,
and partly because Wallis held back the complete solution after it was in his possession. Then, too, they
always communicated through third parties. Fermat
did not understand English, nor Wallis French. In the
meanwhile Fermat wrote several letters to Frenicle
renewing his demand for integers, and in Frenicle's
letter1 to Digby, which reached Wallis February 20,
1658, it was stated that Fermat had solved the equation
x2 - Ay2 = 1
for all non-square values of A up to 150, and that perhaps Wallis would be good enough to extend it to A = 200
or at least solve it for 151, although probably the case
of A = 313 would be beyond his ability. To this last
demand Brouncker replies in a letter2 to Digby which
is dated March 13, 165k, in which he seems to wish it
understood that the solution required but little time.
Using his language and symbolism: "With the space
of an hour or two at most this morning, according to
the method therein delivered, I found that
313 X Q7,170,685 - 1 = Q126,862,368,
and therefore that
313 X Q(2 X 7,170,685 X 126,862,368 =) 1,819,380,158,564,160 + 1 = Q32,188,120,829,134,849,
which I thought fit to present to you that M. Frenicle
may thence perceive that nothing is wanting to the
perfect solution of that problem."
1 "Commercium epistolicum," XXVI, p. 821. "Oeuvres de Fermat,"
vol. III, p. 532.
2 "Commercium epistolicum," XXVII, p. 823. "Oeuvres de Fermat,"
vol. III, p. 536.