Category Archives: Number Theory
When is the group of units in the integers modulo n cyclic?
It is easy to see with the help of Bezout’s identity that the integers co-prime to $n$ from the set $\{0,1,\cdots, n-1\}$ form a group under multiplication modulo $n$. This group is denoted by $(\mathbb{Z}/n\mathbb{Z})^*$ and it’s order is given … Continue reading
Weyl’s Equidistribution Theorem
In this post, we will prove the Weyl’s Equidistribution theorem. A sequence of real numbers $x_1, x_2, \cdots$ is said to be equidistributed (mod 1) if for every sub-interval $(a,b)\subset [0,1]$, we have $$\lim_{N\to \infty}\frac{|\{1\leq n\leq N:\; \langle x_n \rangle\in … Continue reading
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Eisenstein’s Proof of Quadratic Reciprocity
In this post, we present an interesting proof of the Quadratic Reciprocity theorem given by Gotthold Eisenstein. Continue reading
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