abcTRIPLESA,B,Conlyinmathematicscouldsucha.PDFVIP

TRIPLES GREG MARTIN AND WINNIE MIAO ABSTRACT. The conjecture, one of the most famous open problems in number theory, claims that three positive integers satisfying cannot simultaneously have significant repetition among their prime factors; in particular, the product of the distinct primes dividing the three integers should never be much less than . Triples of numbers satisfying are called triples if the product of their distinct prime divisors is strictly less than . We catalog what is known about triples, both numerical examples found through computation and infinite familes of examples established theoretically. In addition, we collect motivations and heuristics supporting the con- jecture, as well as some of its refinements and generalizations, and we describe the state-of-the-art progress towards establishing the conjecture. 1. INTRODUCTION A, B, C . . . only in mathematics could such a trite trio of letters signify a major outstanding open problem with significant connections to multiple topics. The conjecture is a simple-to- state yet challenging problem in number theory that has stumped mathematicians for the past 30 years. It has become known for its large number of profound implications in number theory and particularly in Diophantine equations; among these myriad consequences are Fermat’s last theorem (up to finitely many counterexamples), Mordell’s conjecture [12], and Roth’s theorem [5] (see [29] for a more comprehensive list). The conjecture is deeply intriguing because it unveils some delicate tension between the additive and multiplicative properties of integers, the bread and butter of number theorists. The purpose of this article is to discuss examples and constructions of triples, which a