Maths Olympiad Prep

Library / /285 of 520

Number theory Difficulty 6.1 National olympiad Prove it

18. Let the integer N⩾2N \geqslant 2. Prove:
(i) ∑p⩽N1p−1>ln⁡ln⁡(N+1)\sum_{p \leqslant N} \frac{1}{p-1}>\ln \ln (N+1), where the summation is over all primes pp not exceeding NN;
(ii) ∑p⩽N1p>ln⁡ln⁡(N+1)−1\sum_{p \leqslant N} \frac{1}{p}>\ln \ln (N+1)-1.

Solution

18. (i) Use 1/n>ln⁡(1+1/n)1 / n>\ln (1+1 / n) and
ln⁡(1−1/p)−1=ln⁡(1+1/(p−1))<1/(p−1)\ln (1-1 / p)^{-1}=\ln (1+1 /(p-1))<1 /(p-1)
(ii) ∑p⩽N{1/(p−1)−1/p}<1\sum_{p \leqslant N}\{1 /(p-1)-1 / p\}<1.

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.