Let [θ] denote the nearest integer to θ. i.e. [θ] = ⌊θ + 1/2⌋.
Define a sequence of complete convergents by x0 = (u + √d) / v and xn = bn + an+1 / xn+1, where bn = [xn], n ≥ 0; also an = ±1, where sign(an+1) = sign(xn – bn). This gives a continued fraction x0 = b0 + a1 / b1+ ···, where bi ≥ 2 for all i ≥ 1 and ai = ±1.
Write xk = (Pk + √d) / Qk, so that P0 = u and Q0 = v. Then
bk = [(Pk + √d) / Qk],
Pk+1 = bkQk – Pk,
Qk+1 = ak+1(d – P2k+1) / Qk > 0.
(We use a nearest-integer formula for [x/m], m a non-zero integer.)
We then find the least k ≥ 1 such that Pn+k = Pn and Qn+k = Qn and this leads to a period of length k.
The convergents An / Bn are defined by A-2 = 0, A-1 = 1, B-2 = 1, B-1 = 0 and for k ≥ -1,
Ak+1 = bk+1Ak + ak+1Ak-1
Bk+1 = bk+1Bk + ak+1Bk-1.
If xn = (Pn + √d)/Qn and yn = (Pn - √d)/Qn and r = (3 – √5)/2, then
xn > 2 and -1 + r < yn < r or
xn = 3 – r = (3 + √5)/2.
We then find the least k ≥ 1 such that Pn+k = Pn and Qn+k = Qn and this leads to a period of length k.
The period is printed in bold font.
(11th January 2008. It is not hard to prove that a reduced NICF complete quotient gives a purely periodic NICF and conversely.
The nearest integer continued fraction of Hurwitz and Minnegerode is closely related to the present one, which is described in Perron's Band 1, page 143. This is spelled out in a paper of the author.
Last modified 21st July 2015
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