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

15 votes

A little number theoretic game

Edit: I added code to compute the Nim values for the first $N$ positions of this game after the original post, as requested by @Timothy-Chow. Unfortunately my results don't match those given by @Peter …
I. J. Kennedy's user avatar
46 votes
7 answers
10k views

Are some numbers more irrational than others?

Some irrational numbers are transcendental, which makes them in some sense "more irrational" than algebraic numbers. There are also numbers, such as the golden ratio $\varphi$, which are poorly approx …
I. J. Kennedy's user avatar
28 votes
2 answers
5k views

Is Furstenberg's topology useful?

It's hard not to be amused and perhaps even amazed when first encountering Furstenberg's clever "topological" proof that there are infinitely many primes. Closer inspection, however, reveals the disap …
239 votes
14 answers
76k views

Have any long-suspected irrational numbers turned out to be rational?

The history of proving numbers irrational is full of interesting stories, from the ancient proofs for $\sqrt{2}$, to Lambert's irrationality proof for $\pi$, to Roger Apéry's surprise demonstration th …