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Compute the convergents of the simple continued fraction [0

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HOMEWORK 9

MATH-UA 0248-001 THEORY OF NUMBERS due on Nov, 30, 2020

1. Express the rational number 7155 as a finite simple continued fraction.

2. Compute the convergents of the simple continued fraction [0; 2,4,1,8,2].

3. IfCk = pqk

k is thek-th convergent of the simple continued fraction[a0;a2, . . . an], show thatqk ≥2(k−1)/2. (Hint: Observe that qk =akqk−1+qk−2 ≥2qk−2.) 4. For any positive integer n, show that √

n2 + 1 = [n; 2n]. (Hint: notice that n+√

n2+ 1 = 2n+ (√

n2+ 1−n) = 2n+ 1 n+√

n2+ 1.)

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