Today we shall look at a fairly standard type of argument concerning prime divisors of integer sequences. Given a sequence $latex (a_n)_{n\ge 1}$, we say a prime $latex p$ is a prime divisor of $latex (a_n)$ if $latex p$ is a factor of some term of the sequence. The set of all prime divisors of… Continue reading Prime Supports
Category: Number Theory
Cyclic Digits
Today we shall have a look at a neat little trick that is more commonly seen in computational contests, but occasionally makes an appearance in Olympiad problems as well.Let $latex X=\overline{d_{n-1}d_{n-2}\dots d_0}_b= \sum_{i=0}^{n-1} d_ib^i$ be an integer in base-$latex b$ where $latex 0 \le d_i \le b-1$ for all $latex 0 \le i \le n-1$,… Continue reading Cyclic Digits
Farey Sequences
Today we shall look at a class of sequences with a rich and deep history beginning from their earliest appearance in a problem that appeared in the Ladies' Diary, which have numerous applications in number theory and combinatorics, and are lately making an appearance in math Olympiad problems. The Farey sequence of order $latex n$, denoted $latex \mathcal{F}_n$ is… Continue reading Farey Sequences
Angular Momentum 