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The theory of error-correcting codes stems from Shannon's 1948 _A mathematical theory of communication_, and from Hamming's 1950 "Error detecting and error correcting codes".

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Is there a Kolmogorov complexity proof of the prime number theorem?

Firstly, the proof given there doesn't really show that $p_n = O(n(\log n)^2)$ (at least not without further effort). Instead what it shows is that there are $n$ for which $p_n = O(n(\log n)^2)$, a …
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