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A7

Part of 1958 November Putnam

Problems(1)

Putnam 1958 November A7

Source: Putnam 1958 November

7/19/2022
Let aa and bb be relatively prime positive integers, bb even. For each positive integer qq, let p=p(q)p=p(q) be chosen so that ∣pq−ab∣ \left| \frac{p}{q} - \frac{a}{b} \right| is a minimum. Prove that lim⁡n→∞∑q=1nq∣pq−ab∣n=14. \lim_{n \to \infty} \sum_{q=1 }^{n} \frac{ q\left| \frac{p}{q} - \frac{a}{b} \right|}{n} = \frac{1}{4}.
Putnamnumber theoryrelatively prime