CHAPTER XII.
PELL EQUATION; axz + bx + c MADE A SQUARE.
The very important equation x2 — Dy2 = 1, which has long borne the name of Pell, due to a confusion originating with Euler, should have been designated as Fermat's equation (cf. papers 41, 62-64).
There appeared in India and Greece as early as 400 B.C. approximations aft to V2 such that a2 — 262 = 1, and similarly for other square roots, the derivation of successive approximations being in effect a method of solving the Pell equation. For example, Baudhayana, the Hindu author of the oldest of the works, Sulva-sutras, gave the approximations 17/12 and 577/408 to V2. Note that
17, ~1 12~r2>17-12
577 408'
172 - 2-122 = 1, 5772 - 2-4082 = 1.
Proclus1 (410-485 A.D.) noted that the Pythagoreans made the following construction: On the prolongation of the side AB of a square with the diagonal BE lay off BC = AB, CD = BE. Then
AD2 + CD2
But CD2 = BE2 = 2AB2. Hence AD2 = 2BD2 = FD2,
> 2AB2 + 2BD2.
FD = AD = 2AB + EB.
Also BD = AB + EB. Write slt s2, di, dz, • • • for the diagonals BE, FD, •
= sn
for the sides AB, BD, Then
-, and
dn, dn+i = 2sn + dn>
Now let $i = 1 and replace di = A/2 by the integral approximation 5i = 1, and employ our recursion formulae with dn replaced by 5n. We get
52 = 81 + 5i = 2, S2 = 28! + Ği = 3,
53 = s2 + S2 = 5, 5s = 2s2 + 52 = 7, Then 6n, sn give a solution of S2 — 2s2 = (— l)n.
1 In Platonis rem publicam commentarii, ed., G. Kroll, 2, 1901, 2^-9; excurs II (by F. Hultsch), 393-400.
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