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Difference of j-invariant values and the abc conjecture

I learned of the following example in a recent seminar: if $j(\tau)$ denotes the usual $j$-invariant, and $\alpha = (-1+i\sqrt{163})/2$, then \begin{align*} \frac{j(i)}{1728} &= 1 \\ \frac{-j(\...
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Have there been any updates on Mochizuki's proposed proof of the abc conjecture?

In April 2012, a proof of the abc conjecture was proposed by Shinichi Mochizuki. However, the proof was based on a "Inter-universal Teichmüller theory" which Mochizuki himself pioneered. It was known ...
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Estimate on radical of $2^n \pm 1$

Not sure if this belongs to MO or not. Are there any lower bound on radical of $2^n \pm 1$? We recall that radical of an integer $rad(k)$ is a product of primes which divide $k$. As an example, if ...
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A converse of the abc conjecture?

Let ${\rm rad}(n)$ denote the radical of a positive integer $n$, i.e. the product of its distinct prime divisors. Given positive integers $a$ and $b$, the triple $(a,b,a+b)$ is called an abc triple if ...
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Does Mochizuki's proof of abc conjecture gives an upper bound for the quality of a triple?

The quality of a triple $(a,b,c)$ of coprime positive integers with $a + b = c$ is defined as $$q(a,b,c) := \frac{\log(c)}{\log(\mathrm{rad}(abc))}.$$ Then $$a+b = c = \mathrm{rad}(abc)^{q(a,b,c)}.$$...
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Probing the generalization of the abc conjecture to more than 3 variables

Browkin and Brzezinski, in "Some remarks on the $abc$-conjecture", Math. Comp. 62 (1994), no. 206, 931–939, state the following generalization of the $abc$ conjecture to more than three variables: ...
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Model-theoretic content of Mochizuki's Teichmüller theory papers

I would like to ask what the specific novel model-theoretic (or set-theoretic) techniques, if any, are that Mochizuki uses in his recent series of four papers. Section 3 of Inter-universal Teichmüller ...
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