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3 votes
1 answer
954 views

A geometric proof that there are infinitely many primes?

Let $e_d$ be the $d$-th standard-basis vector in the Hilbert space $H=l_2(\mathbb{N})$. Let $h(n) = J_2(n)$ be the second Jordan totient function, defined by: $$J_2(n) = n^2 \prod_{p|n}(1-1/p^2)$$ ...
24 votes
4 answers
2k views

A reinterpretation of the $abc$ - conjecture in terms of metric spaces?

I hope it is appropriate to ask this question here: One formulation of the abc-conjecture is $$ c < \text{rad}(abc)^2$$ where $\gcd(a,b)=1$ and $c=a+b$. This is equivalent to ($a,b$ being ...
9 votes
0 answers
365 views

How to count integer lattice points close to a subspace of $\mathbb R^n$?

Consider $m$ linearly independent vectors in $n$-dimensional Euclidean space, $v_1,...,v_m \in \mathbb R^n$ where $1\leq m<n$, and let $U := {\rm span}(v_1,...,v_m)$ denote the $m$-dimensional ...