Linked Questions

7
votes
0answers
248 views

Proving infinitely many primes using algebraic geometry ideas

There are at least two well known proofs of the infinitude of primes (Euclid's original one and Euler's proof using L-series) and both of them can be extended to prove more general statements of the ...
9
votes
5answers
479 views

Reference request: Recovering a Riemannian metric from the distance function

Let $M = (M, g)$ be a Riemannian manifold, and let $p \in M$. Writing $d$ for the geodesic distance in $M$, there is a function $$ d(-, p)^2 : M \to \mathbb{R}. $$ This function is smooth near $p$. ...
8
votes
1answer
461 views

Are the following subsets of a Hilbert space always homeomorphic?

Let $F$ be a infinite-dimensional complex Hilbert space, with inner product $\langle\cdot\;| \;\cdot\rangle$, the norm $\|\cdot\|$, the 1-sphere $S(0,1)=\{x\in F;\;\|x\|=1\}$ and let $\mathcal{B}(F)$ ...
4
votes
1answer
671 views

Use of infinitude of primes in the Green-Tao theorem [closed]

In a video I watched last night on nuking mathematical mosquitos, Matt Parker gave the following proof of the infinitude of primes: suppose there are finitely many primes. The Green-Tao theorem says ...
3
votes
4answers
2k views

Non-separable Banach space

The vector space $C_b(\mathbb R)$ of bounded continuous functions on $\mathbb R$ is non-separable: it is possible to produce a direct proof of this fact, mimicking the standard proof for the non-...
53
votes
4answers
6k views

Has Fermat's Last Theorem per se been used?

There is a long tradition of mathematicians remarking that FLT in itself is a rather isolated claim, attractive only because of its simplicity. And people often note a great thing about current ...
5
votes
3answers
1k views

When is the graph of a function a dense set?

Let $f: \mathbb R \to \mathbb R$ be any function. When is the graph of $f$ dense in $\mathbb R^2$? The only examples I know for this are for non-measurable functions, but is that a necessary condition?...
21
votes
10answers
3k views

Completeness vs Compactness in logic

One standard approach to showing compactness of first-order logic is to show completeness, of which compactness is an easy corollary. I am told this approach is deprecated nowadays, as Compactness is ...
184
votes
64answers
34k views

Proofs that require fundamentally new ways of thinking

I do not know exactly how to characterize the class of proofs that interests me, so let me give some examples and say why I would be interested in more. Perhaps what the examples have in common is ...
5
votes
1answer
3k views

Has Stirling’s Formula ever been applied, with interesting consequence, to Wilson’s Theorem?

Pressing the envelope, presumably the best scenario would be a simple proof of the Prime Number Theorem. After all, Wilson’s Theorem gives a necessary and sufficient condition, in terms of the Gamma ...
54
votes
4answers
12k views

Proof that pi is transcendental that doesn't use the infinitude of primes

I just taught the classical impossible constructions for the first time, and in finding my class a reference for the transcendence of pi, I found a dearth of distinct proofs. In particular, those ...
38
votes
4answers
5k views

Is there an “elementary” proof of the infinitude of completely split primes?

Let $K$ be a Galois extension of the rationals with degree $n$. The Chebotarev Density Theorem guarantees that the rational primes that split completely in $K$ have density $1/n$ and thus there are ...