Number theory is a branch of pure mathematics devoted primarily to the study of the natural numbers and the integers. It is especially concerned with studying prime numbers and the properties of integers and objects made from them (e.g. rational numbers). The history of this discipline is fascinating and replete with astonishingly clever mathematicians and famous theorems and insights. You will learn of Diophantine equations, the Riemann zeta function, and many other amazing things.

This was a field of study that fascinated me as a school student (and beyond). I spent many happy hours analysing primes, sequences, the density of primes, divisibility, etc. Until I add material here, let me recommend that you follow the link to the Wikipedia article about Number Theory and then simply Google the term and browse.

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