The Pell equation, by Edward Everett Whitford.

THE PELL EQUATION

77

If the form AX2 + 2BXY + CY2 = F implies the form
ax2 + 2bxy + cy2 = f, and if a certain transformation is
given which transforms the first into the second; from
this to deduce all the transformations which produce the
same result.
The solution is substantially as follows: If the given
transformation is X = ax + py and Y = yx + by, we first
let another similar transformation produce the same
result. Let this new transformation be X = a'x + b'y,
Y = y'x + 6'y. Let the determinants of the forms F and
f be D and d, and let ca -,y = e, a'8' - '^y' = e'.
Then d = De2 = De'2, and since by hypothesis e and e'
have the same sign, it follows that e = e'. We have
moreover the following six equations:
(1) Aa2 + 2Bay + Cy2 = a,
(2) Aa'2 + 2Ba'y' + Cy'2 = a,
(3) AaO + B(a5 + y7) + Cya = b,
(4) Aa'/' + B(a'a' + 3'y') + Cy'a' = b,
(5) A32 + 2B13 + C2 = c,
(6) Af3'2 + 2Bf3'a' + C'2 = c.
If for the sake of brevity we designate the numbers
Aaa' + B(ay' + a'y) + Cyy', A(aO3' + Oa') + B(a3' +
Oy' y' + 6a') + C(yb' + 5y'), A33' + B(/6' + 53')
+ C6S' by a', 2b', c', respectively, we deduce from the
preceding equations the following:
(7) a'2- -D(ac' - ya')2 = a2
(8) 2a'b' - D(ay' - ya')(a/ + ry' - ' - 5a') = 2ab,
4b'2 - D[(a/5' +,y' - 7/3' - 6a')2 - 2ee'] = 2b2 + 2ac.
By adding 2Dee' = 2d = 2b2 - 2ac to the equation above
we have

(9) 4b'2 - D(aS' + y,' - 7y3' - ba')2 = 4b2,