Positional notation. An arbitrary precision integer on a computer is represented as a list of digits (limbs). They are just bigger. Today’s computers can hold 64 bits per limb. Modular arithmetic consists in applying addition and multiplication to integers modulo a certain number \(n\). Given an implementation of addition, multiplication, and division on the integers, we can emulate addition or multiplication modulo \(n\) by performing the corresponding operation on integers then taking the result modulo \(n\).

The erroneous proof I hear most often is: Suppose \(P\) is a finite set that contains all the primes, then \(p^* = 1 + \prod_{p \in P} p\) is prime. Indeed, the flaw is that \(p^*\) is not necessarily prime but rather must be a multiple of some prime not in \(P\).