I will talk about a recent work of 5 coauthors: Sh. Mochizuki, W. Porowski, A. Minamide, Yu. Hoshi and I. This work slightly extends the IUT theory of Shinichi Mochizuki (for an updated short description of the study of IUT see https://www.maths.nottingham.ac.uk/plp/pmzibf/rapg.pdf). It incorporates the residue characteristic at $p=2$. Using computations of Sijsling (2019) of $4$ special cases of $j$-invariants, it then produces effective estimates of constants. This leads to the proof of effective form of one of $abc$ inequalities. In applications of this form of $abc$ inequality to diophantine equations one can use two additional tools: bounds from below on their solutions and some computer verifications. This opens a vast area of further developments. In the particular case of FLT, using bounds from below obtained by Inkeri (1987) and computations by Coppersmith (1990) and Hart-Harvey-Ong (2016), this recent work proves the first case of FLT for all prime exponents and the second case of FLT for all prime exponents except those between $2^{31}$ and $9.6\times 10^{13}$.