## The abc Theorem

In 2012 Shinichi Mochizuki shared his four preprints proving the abc conjecture, which is concerned with number theory. In 2020 these manuscripts were accepted for publication, and the historical papers published in 2021:

Publications of the Research Institute for Mathematical Sciences, Volume 57, Number 1/2, 2021, Special Issue, url.

We introduce some notions and notation to state the abc theorem.
• Suppose that the prime factorization of a positive integer $$n$$ is $$n=p_1^{m(1)} \dotsb p_l^{m(l)}$$. The radical of $$n$$ is defined by $$\operatorname{rad}(n):=p_1 \dotsb p_l$$.
• We denote by $$X$$ the set of triples $$(a,b,c)$$ of positive integers such that $a < b, \quad a+b=c, \quad \operatorname{gcd}(a,b)=\operatorname{gcd}(b,c)=\operatorname{gcd}(c,a)=1.$
• For $$\kappa \geqq1$$, $$X[\kappa]:=\{(a,b,c) \in X : c \geqq \operatorname{rad}(abc)^\kappa\}$$.
The abc theorem asserts that
$$X[\kappa]$$ is a finite set for any $$\kappa>1$$.
We remark that
• If $$\kappa>\mu$$, then $$X[\kappa] \subset X[\mu]$$.
• $$X[1]$$ is an infinite set. Indeed $$\{(1,3^{2^n}-1,3^{2^n})\}_{n=1}^\infty \subset X[1]$$.
• The only three elements of $$X[1.6]$$ were discovered so far: $(2,3^{10}\cdot109,23^5), \quad (11^2,3^2\cdot5^6\cdot7^3,2^{21}\cdot23), \quad (19\cdot1307,29^2\cdot31^8,2^8\cdot3^{22}\cdot5^4).$
We find some examples of tiriples $$(a,b,c) \in X[\kappa]$$ for $$\kappa=1.4$$.

 Examples of elements of $$X[1.4]$$ abc.m   acb14.xlsx