abc Conjecture

th

n

coprime

13

a + b = c: 239 +

8

5

·

3

17

=

10

2

·

4

37

rad(a, b, c) = rad(

10

2

·

8

5

·

3

17

·

4

37

·239) = 1503310 < c

rad(a, b, c) / c =

20315

25934336

= 0.000783324 < 1

q(a, b, c) = q(239, 1919140625, 1919140864) = 1.50284 > 1

The

abc

conjecture involves triples

a

,

b

,

c

of relatively prime positive integers satisfying

a+b=c

. The conjecture was first proposed by Joseph Oesterle in 1985 and David Masser in 1988. The function

rad(abc)

(the radical) is the product of the distinct prime factors of the triple (the square-free part of

abc

). The conjecture states that this product is "rarely" much smaller than

c

: there are finitely many such exceptions. For a typical triple, the "quality" of the triple,

q(a,b,c)=log(c)/log(rad(abc))

, is less than 1; for exceptions,

q(a,b,c)>1+ε

, with

ε

small. Values above 1.4 are rare; in fact, only 234 are known [2], so they are considered "high quality triples".

This conjecture is one of the most important open problems in Diophantine analysis, encoding profound connections among the prime factors of the triples as well as connections with already-proved theorems, including Fermat’s last theorem. If proved, it would imply other conjectures.