103 missions · 51 completed
The study of the integers and the structures built from them — prime numbers, and the rational, algebraic, and -adic numbers that extend them. It reaches from analytic number theory, which uses the tools of analysis to understand the distribution of primes, to algebraic number theory, Diophantine equations, and the arithmetic of elliptic curves, modular forms, and -functions.
Erdős Problem 788 asks how large a set can always be retained when prescribed distinct pair-sums are forbidden. This mission formalizes the repository’s strengthened version of Theorem 1.1: an explicit lower bound valid for every , an eventual quantitative upper bound, the conclusion , and the exact affirmative answer to the original upper-bound question.
Among the most enduring mysteries in number theory is whether the primes keep producing twins — pairs like (11, 13) or (17, 19) that differ by exactly two — no matter how far out one looks. The general form was set down by Alphonse de Polignac in 1849, and the first deep theorem came from Viggo Brun in 1915, who proved that the reciprocals of the twin primes converge to a finite value, now called Brun's constant; in doing so he invented modern sieve theory and showed that twins must thin out even if there are infinitely many. Hardy and Littlewood went further, conjecturing a precise density of about 2C₂·x/(ln x)² for the count of twins below x. For nearly a century the infinitude itself stood untouched, until Yitang Zhang's stunning announcement on 17 April 2013 that some gap below 70 million recurs infinitely often — the first finite bound ever proved. A Polymath collaboration led by Terence Tao, together with James Maynard's independent multidimensional sieve, soon drove that bound down to 246, where it still stands. Closing the gap all the way to 2 — the twin prime conjecture itself — remains open. This mission states it cleanly: the set of primes p for which p + 2 is also prime is infinite.
Formulated in 1985 by Joseph Oesterlé and David Masser as an arithmetic distillation of Szpiro's conjecture on elliptic curves, the abc conjecture makes a deceptively simple claim about coprime triples with a + b = c: the three numbers cannot all be built from many repeated small primes at once, so c can only rarely exceed rad(abc)^(1+ε). Dorian Goldfeld called it 'the most important unsolved problem in Diophantine analysis,' and for good reason — a single proof would cascade through number theory, delivering Fermat's Last Theorem for all large exponents almost for free, along with Roth's theorem, the Mordell–Faltings theorem, the Fermat–Catalan conjecture, infinitely many non-Wieferich primes, and all but finitely many counterexamples to Beal's conjecture. Since 2012 Shinichi Mochizuki has claimed a proof via inter-universal Teichmüller theory, published in 2021, but the community has not accepted it: in 2018 Peter Scholze and Jakob Stix identified a gap they regarded as fatal. A precise formal statement gives everyone a shared, machine-checkable target around which to organize verified progress.