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Show by contradiction that
$$
\sum_{p\in \mathbb{P}}\dfrac{1}{p} =\sum_{p\;\text{prime}}\dfrac{1}{p}
$$
diverges. Which famous result is an immediate corollary?
$$
\sum_{p\in \mathbb{P}}\dfrac{1}{p} =\sum_{p\;\text{prime}}\dfrac{1}{p}
$$
diverges. Which famous result is an immediate corollary?
Skeleton (proof by Pál Erdös):
- $$\underbrace{p_1<\ldots <p_k<}_{\text{small primes}}\underbrace{\underbrace{p_{k+1}<\ldots}_{\text{big primes}}}_{\displaystyle{\sum_{j>k}\dfrac{1}{p_j}<\dfrac{1}{2}}}$$
- Set ##N_b =\# \{n \leq N\,|\,\exists \,j>k\, : \,p_j|n\}## and ##N_s=\#\{n\leq N\,|\,p_j|n\Longrightarrow j\leq k\}.##
- ##N_b < \dfrac{N}{2}##
- ##N_s < 2^k\sqrt{N}##
- ##N_b+N_s < N##
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