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Sum of Reciprocals of Products of Pairs of Primes

Source: Junior Olympiad of Malaysia Shortlist 2015 N6

July 17, 2015
number theory

Problem Statement

Let pi p_i denote the i i -th prime number. Let n=⌊α2015⌋ n = \lfloor\alpha^{2015}\rfloor , where α \alpha is a positive real number such that 2<α<2.7 2 < \alpha < 2.7 . Prove that ∑2≤pi≤pj≤n1pipj<2017 \displaystyle\sum_{2 \le p_i \le p_j \le n}\frac{1}{p_ip_j} < 2017
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