The Pell equation, by Edward Everett Whitford.

18

THE PELL EQUATION

this is clearly an approximation for ala. If we set b2a
in place of a we may thus make a new a that is somewhat
nearer to unity. Thus suppose we have
27
a 25'
so that
3
5
and put
Do = So = 1;
then
52 5 26 26
S = 2, D3 or
1=2, =251' 4 3 25 r15'
102 54 + 52 - 5 106 265
S2 - 25' 25D2- 3 1- - or
S =25 25 ' 3 3 102 153'
208 5404 5 5404 1351
25' D325 25' 3 25 208 r 780;
the two Archimedian approximations coming close together. Other methods are given by Hunrath, Hultsch,
Rodet, Oppermann, Sch6nborn, many of the solutions
depending on the relation
b b
a 2a > a2 + b > a i 2a -- 1
where a2 + b is a non-square integer and a2 is the nearest
square number to it.
If in the general equation
X2- ay2 = b
we have found by trial the three simplest solutions,
Tannery has shown the law of forming successive solutions.' If
x=p, y =q
1 P. Tannery, "Sur la mesure...," loc. cit. T. L. Heath, " Diophantus
of Alexandria," p. 279, 2d ed., Cambridge, 1910.