Peter Szüsz

PERIODIC CONTINUED FRACTIONS

The goals of this project are to have the reader explore some of the basic properties of continued fractions and prove that α ∈ R is a quadratic irrational iff α is equal to a periodic continued fraction. 1. Finite Continued Fractions Fix s = (a0, (a1, . . . , an)) ∈ Z × N. The finite (simple) continued fraction of s is defined as [s] = [a0; a1, . . . , an] = a0 + 1 a1 + 1 a2 + · · ·+ 1 an , and if n ≥ k ∈ N0, the kth convergent ck of s is taken as ck = [a0; a1, . . . , ak]. Prove that k ∈ {0, . . . , n} ⇒ (1.1) ck = pk qk where p0 = a0, p1 = a0a1 + 1, q0 = 1, q1 = a1, and pm = ampm−1 + pm−2, (1.2) qm = amqm−1 + qm−2 (1.3) for all m ∈ {2, . . . , n}. Next, use 1.1 and the definitions of pm, qm to show that (1.4) ck−1 − ck = (−1) qkqk−1 for all k ∈ {1, . . . , n} (Hint: Subtract qm−1 times 1.2 from pm−1 times 1.3.). Now employ 1.4 to conclude (1.5) c0 ≤ c2 ≤ c4 ≤ . . . ≤ [s] ≤ . . . ≤ c5 ≤ c3 ≤ c1. Also, use 1.4 again to demonstrate that (pk, qk) = 1 for all k ∈ {0, . . . , n}. 2. Infinite Continued Fractions Fix t = (a0, (a1, a2, . . .)) ∈ Z × N∞ and extend the definitions of ck, pk, qk to all k ∈ N0. Prove that the limit (called the infinite (simple) continued fraction of t) lim k→∞ ck = [t] = [a0; a1, a2, . . .] = a0 + 1