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, xn = an – 1/xn+1, where an = [xn], n ≥ 0.
This gives a continued fraction x0 = a0 – 1/a1– ···, where |ai| ≥ 2 for all i ≥ 1.
The notation x0 = (a0,a1,...) goes back to A. Hurwitz, Werke, Band II, Seite 85.
Write xk = (Pk + √d)/Qk, so that P0 = u and Q0 = v. Then
qk = [(Pk + √d)/Qk],
Pk+1 = qkQk – Pk,
Qk+1 = (P2k+1 – d)/ Qk.
(We use a nearest-integer formula for [x/m], m a non-zero integer.)
The first Hurwitz-reduced complete quotient xn = (Pn + √d)/Qn (see Werke, Band II, Seite 102) is located. ie.
with yn = (Pn – √d)/Qn and r = (3 – √5)/2,
xn > 2 and -1 + r < yn < r or
xn < -2 and -r < yn < 1 – r or
xn = 3 – r = (3 + √5)/2, 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 convergents An/Bn are defined by A-2 = 0, A-1 = 1, B-2 = -1, B-1 = 0 and for k ≥ -1,
Ak+1 = qk+1Ak – Ak-1
Bk+1 = qk+1Bk – Bk-1.
The nearest integer continued fraction of Hurwitz and Minnegerode is closely related to the one in Perron's Band 1, page 143. This connection is spelled out in a paper of the author.
Last modified 21st July 2015
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