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Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions

36 votes
8 answers
32k views

What are the connections between pi and prime numbers?

I watched a video that said the probability for Gaussian integers to be relatively prime is an expression in $\pi$, and I also know about $\zeta(2) = \pi^2/6$ but I am wondering what are more connecti …
15 votes

Work on independence of pi and e

There is a proof of the algebraic independence of $\pi$ and $e^\pi$ in Introduction to Algebraic Independence Theory and a detailed exposition of methods created in last the 25 years although I have n …
muad's user avatar
  • 1,412
33 votes
3 answers
8k views

Work on independence of pi and e

It is an open problem to prove that $\pi$ and $e$ are algebraically independent over $\mathbb{Q}$. What are some of the important results leading toward proving this? What are the most promising the …
muad's user avatar
  • 1,412