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 \(\kappa \geqq1\), \(X[\kappa]:=\{(a,b,c) \in X : c \geqq \operatorname{rad}(abc)^\kappa\}\).

- 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: \[ (2,3^{10}\cdot109,23^5), \quad (11^2,3^2\cdot5^6\cdot7^3,2^{21}\cdot23), \quad (19\cdot1307,29^2\cdot31^8,2^8\cdot3^{22}\cdot5^4). \]