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2017 T7: Rational Approximation of sqrt(2)

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January 29, 2017
2017team

Problem Statement

Define {pn}n=0∞⊂N\{p_n\}_{n=0}^\infty\subset\mathbb N and {qn}n=0∞⊂N\{q_n\}_{n=0}^\infty\subset\mathbb N to be sequences of natural numbers as follows:
[*]p0=q0=1p_0=q_0=1; [*]For all n∈Nn\in\mathbb N, qnq_n is the smallest natural number such that there exists a natural number pnp_n with gcd⁡(pn,qn)=1\gcd(p_n,q_n)=1 satisfying pn−1qn−1<pnqn<2.\dfrac{p_{n-1}}{q_{n-1}} < \dfrac{p_n}{q_n} < \sqrt 2.
Find q3q_3.
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