## Topic

The Hunt for ABC Triples

## Speaker

Frank Rubin

## When

**Monday,
7:30 PM**

## Where

Marist College,
**Hancock Center** (Building 14 on
map), **Room 2023**.
We thank Marist College for hosting the chapter's meetings.
**Parking:** 1) Lot north of Hancock Center.
We are permitted to park there in spite of the signs.
2) Lot on south-east corner of Route 9 and Fulton Street.
Take staircase at south-west corner of lot, and tunnel under Route 9.

## More Information

This program is **free**
and **open to the public**.
Attendees should RSVP at
Meetup.com.

All are welcome to join us beforehand for **dinner** at the
Palace Diner at **6:00 PM**.

For further information, go to **Pok.ACM.org** (QR code below):

**ACM Poughkeepsie**at

## About the Topic

A triple A, B, C of positive integers with no common divisor d > 1 is called an *ABC Triple*
if A + B = C, and the product of the distinct prime factors of ABC is less than C.
Two well-known examples are 2^{2} + 11^{2} = 5^{3}
and 3 + 5^{3} = 2^{7}.
The product of the distinct prime factors of an integer N is called the *radical* of N,
denoted rad(N).
So rad(6) = rad(12) = rad(18) = rad(144) = 6.
Since ABC is the product of 3 integers, one of which is C itself,
rad(ABC) cannot be smaller than C unless C has several repeated prime factors,
and A and B either have repeated prime factors or they are much smaller than C.

There is a direct connection between ABC Triples and Fermat's Last Theorem,
because if A^{n} + B^{n} = C^{n}
with n > 2, then A^{n}, B^{n}, C^{n} would be an ABC Triple.
The theorem states that no such triple exists.
Let R = rad(ABC).
The *quality* Q of the triple is defined as Q = ln(C)/ln(R).
When Q > 1.4 the triple is called *good*.
Good triples up to 30 digits are known.
The ABC Conjecture is that for any quality Q' > 1 there are only finitely many ABC Triples with Q > Q'.

The *merit* M of a triple is defined in terms of the quality
as M = (Q - 1)^{2} ln(R) ln(ln(R)).
Triples with merit up to 38.67 are known.
It is believed that merit can never exceed 48.
Triples with M > 24 are called *high merit*.
High merit triples up to 253 digits are known.
A triple D + E = F is said to *beat* a triple A + B = C if F > C and D, E, F has higher quality
than A, B, C.
If there is no triple that beats A, B, C, then A,B,C is called *unbeaten*.
Unbeaten triples up to 13,331 digits are known.

This talk will describe computer algorithms for finding high quality, high merit and unbeaten triples.

## About the Speaker

Frank Rubin has an M.S. in Mathematics and a Ph.D. in Computer Science. He has been hunting for ABC Triples since 2007, finding most of the good triples over 25 digits, more high merit triples than the next 2 searchers combined, and all of the unbeaten triples over 3000 digits.

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