Functiones et Approximatio Commentarii Mathematici

$abc$ triples

Greg Martin and Winnie Miao

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The $abc$ conjecture, one of the most famous open problems in number theory, claims that three relatively prime positive integers $a,b,c$ satisfying $a+b=c$ 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 $c$. Triples of relatively prime numbers satisfying $a+b=c$ are called {\em $abc$ triples} if the product of their distinct prime divisors is strictly less than $c$. We catalog what is known about $abc$ triples, both numerical examples found through computation and infinite familes of examples established theoretically. In addition, we collect motivations and heuristics supporting the $abc$ conjecture, as well as some of its refinements and generalizations, and we describe the state-of-the-art progress towards establishing the conjecture.

Article information

Funct. Approx. Comment. Math., Volume 55, Number 2 (2016), 145-176.

First available in Project Euclid: 17 December 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11N32: Primes represented by polynomials; other multiplicative structure of polynomial values 11N25: Distribution of integers with specified multiplicative constraints 11N05: Distribution of primes
Secondary: 11R29: Class numbers, class groups, discriminants

$abc$ conjecture number theory factorization


Martin, Greg; Miao, Winnie. $abc$ triples. Funct. Approx. Comment. Math. 55 (2016), no. 2, 145--176. doi:10.7169/facm/2016.55.2.2.

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