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Over the history of mathematics, some key facts have had multiple and different proofs developed for them. Sometimes these different proofs provide a unique insight or understanding of those facts. Fo …

I am reading several sources and there seems to be a lack of clarity, and some contradiction, about the origins of the most recognised prime counting function approximations: $\pi(n) \sim \frac{n}{\ …

Gauss, in his 1849 letter to Encke, mentions that he noticed the primes have a density approx $\frac{1}{\ln(n)}$. In that letter, he also mentions an integral function for approximating the prime co …

Euler developed the Euler Product Formula which shows that the Riemann zeta function encodes information about the prime. $$\zeta(s)=\sum_{n}\frac{1}{n^{s}}=\prod_{p}(1-\frac{1}{p^{s}})^{-1}$$ Riemann …

The Cramér probabilistic model of primes is built on the assumption that the probability of $n$ being prime is $$\Pr(n)=\frac{1}{\log (n)}$$ This is not a big leap from the Prime Number Theorem which …

I am looking to see whether historic mathematicians did any work to explain the slightly unexpected distribution of prime gaps? I would have expected Gauss, who studied lists of primes and proposed a …

When and where did Legendre first publish or write about his conjecture that there is a prime between consecutive square numbers? $$n^2 < p < (n+1)^2$$ I have looked through edition 1 and 2 of his E …

Q. Is it true that Euler's proof of infinite primes was the first since Euclid's which was from around 300BC? Note: By Euler's proof, I mean the use of his Euler product formula for the zeta function …