In 1988, Masser and Oesterlé conjectured that if A,B,C are co-prime integers satisfying

A + B = C,

then for any ε > 0,

max{|A|,|B|,|C|}≤ K(ε)Rad(ABC)[superscript]1+ε,

where Rad(n) denotes the product of the distinct primes dividing n. This is known as the

ABC Conjecture. Versions with the ε dependence made explicit have also been conjectured.

For example in 2004 A. Baker suggested that

max{|A|,|B|,|C|}≤6/5Rad(ABC) (logRad(ABC))ω [over] ω!

where ω = ω(ABC), denotes the number of distinct primes dividing A, B, and C. For

example this would lead to

max{|A|,|B|,|C|} < Rad(ABC)[superscript]7/4.

The ABC Conjecture really is deep. Its truth would have a wide variety of applications

to many diﬀerent aspects in Number Theory, which we will see in this report. These include

Fermat’s Last Theorem, Wieferich Primes, gaps between primes, Erdős-Woods Conjecture,

Roth’s Theorem, Mordell’s Conjecture/Faltings’ Theorem, and Baker’s Theorem to name

a few. For instance, it could be used to prove Fermat’s Last Theorem in only a couple of

lines. That is truly fascinating in the world of Number Theory because it took over 300

years before Andrew Wiles came up with a lengthy proof of Fermat’s Last Theorem.

We are far from proving this conjecture. The best we can do is Stewart and Yu’s 2001

result

max{log|A|,log|B|,log|C|}≤ K(ε)Rad(ABC)[superscript]1/3+ε. (1)

However, a polynomial version was proved by Mason in 1982.