##### Department of Mathematics,

University of California San Diego

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### Math 295 - Mathematics Colloquium

## Hendrik W. Lenstra Jr.

## Searching for good $abc$-triples

##### Abstract:

An *$abc$-triple* is a triple of pairwise coprime positive integers $a$, $b$, $c$ with $a$ + $b$ = $c$. The *radical* $r$ of such a triple is the product of the distinct prime numbers dividing $abc$, and the *quality* $q$ is defined by $q = (log c)/log r$. For example, the triple given by $a = 5, b = 27, c = 32$ has $r = 30$ and $q = (log 32)/log 30 = 1.018975235$... The *$abc$-conjecture* asserts that for any real number $Q > 1$, the number of $abc$-triples with quality greater than $Q$ is finite. It is known that there do exist infinitely many $abc$-triples with quality greater than $1$. The main subject of the lecture is an algorithm for listing, given a large integer $N$, all $abc$-triples with $c$ at most $N$ and quality greater than $1$. As a byproduct, the algorithm yields an upper bound for the number of such triples, as a function of $N$.

Host: J. Buhler

### January 19, 2006

### 3:00 PM

### AP&M 7321

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