The Pell equation, by Edward Everett Whitford.

90 THE PELL EQUATION
As we have
s2. 2 27r 2.
1-e x = -2i.sins2-e P,
p
the last equation takes the form
_ p+l p —1
- P+l P-l 7r 7r
g + hp = 22 (-1) 4 sin 12.- sin 22
sin P — 2 1 [e12+22+...+(2 -1)2]
2 p
But
12 22 ** [P- 1] P2 - 1
2 p 24
where (p2 - 1)/24 is an integer, odd or even according
as p has the form 8n + 1 or 8n + 5. The exponential
factor is either + 1 or - 1 and can be expressed as (- )-l)/ 4.
Substituting,
pw- I /_ l
g+h/p-=2 2 *sin1. sin22-.. sin 2)
p p 2 p
From (5) we see that the integer g is divisible by p.
Putting pk in place of g,
h2- pk2 = - 4,
and
p+1
h + k1p= X sin12 sin22.sin 2
p p 2 p
We see then that there exist integers h and k such that
h2 - pk2 = - 4,
and that these integers can be expressed in general by
circular functions. From the preceding equations
a 2 1 (a 2
=k= —[~ -.
h 2 a k \ 2+ a/
After distinguishing the cases
p = 8n + 1