The Hunt for ABC Triples
Monday, 7:30 PM
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.
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):
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 22 + 112 = 53 and 3 + 53 = 27. 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 An + Bn = Cn with n > 2, then An, Bn, Cn 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.