The purest branch of mathematics ended up guarding online secrets
Carl Friedrich Gauss crowned number theory the queen of mathematics, and for generations it was prized precisely because it seemed useless outside the subject. Then in the 1970s prime numbers became the backbone of public-key cryptography such as RSA, and the study of whole numbers suddenly had very practical stakes.
Number theory, traditionally called higher arithmetic, studies the integers and their properties, including primes, fractions built from integers and generalisations such as algebraic integers. It splits into elementary, analytic, algebraic and geometric branches, among others; the analytic side uses tools like the Riemann zeta function. Its trademark is questions anyone can grasp that resist the best minds. Fermat's Last Theorem took 358 years to prove, and Goldbach's conjecture has stood open since the 18th century.
The oldest arithmetical artefact is Plimpton 322, a broken Babylonian clay tablet from around 1800 BC listing Pythagorean triples too many and too large to find by trial and error. It may have supplied examples for school exercises, and it is the only surviving trace of Babylonian number theory. Greek arithmetic seems to have grown independently. The Pythagoreans saw mystical meaning in perfect and amicable numbers, Euclid gave his algorithm for the greatest common divisor and an argument that primes never end, and Diophantus of Alexandria, probably in the 3rd century AD, compiled problems seeking rational solutions to polynomial equations.
Later progress moved east. The Chinese remainder theorem appears as an exercise in the Sunzi Suanjing, and Qin Jiushao gave a complete general method in 1247. Āryabhaṭa's pulveriser resembled Euclid's algorithm and seems aimed at astronomy, Brahmagupta took up the Pell equation in 628, and Bhāskara II's twelfth-century text preserves a general solution probably due to Jayadeva. In Baghdad the caliph al-Ma'mun sponsored translations, Qusta ibn Luqa rendered Diophantus into Arabic, and Ibn al-Haytham apparently knew what is now Wilson's theorem.
Medieval Western Europe contributed little beyond a treatise by Fibonacci. Pierre de Fermat, who published nothing and scribbled results in margins and letters, revived interest with his little theorem and his famous last one. In 1729 the amateur Christian Goldbach pointed Leonhard Euler to Fermat's work, sparking what has been called the rebirth of the field, and Euler went on to prove many of Fermat's claims.
Source: Number theory
