# Questions tagged [abc-conjecture]

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### Consequences of Kirti Joshi's new preprint about p-adic Teichmüller theory on the validity of IUT and on the ABC conjecture

Today, somebody posted on the nLab a link to Kirti Joshi's preprint on the arXiv from last month: https://arxiv.org/abs/2210.11635 In that preprint, Kirti Joshi claims that he agrees with Scholze and ...
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### Simple Diophantine equation

Are there any solutions in positive integers of $x^3 + 1 = (x - k) y^3$? The closest I can get is $19^3 + 1 = 20 \times 7^3$, but $20\gt 19$ so it just misses! For the related $x^3 - 1 = (x - k) y^3$,...
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### Some statistics related to the abc conjecture

We did some statistics about the 14 million good abc triples below 10^18 taken from Bart de Smith site. This was examining just the top of the iceberg, since the interesting triples grow very likely ...
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### Argument against Vojta's more general abc conjecture

Confusion is possible, we got argument against Vojta's more general abc conjecture. In A more general abc conjecture, p. 7 Paul Vojta conjectures: If $x_0,\ldots x_{n-1}$ are nonzero coprime ...
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### How balanced can abc triples be?

I was looking at the $241$ known "good" abc triples (i.e. with quality $\geqslant1.4$), wondering how frequently $a$ and $b$ would have more or less the same order of magnitude. The outcome is not ...
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### The abc-conjecture over the positive rationals and Levy-Schoenberg kernels?

I am continuing the "abc-adventure" and have a specific question, which needs some explanation: In this paper by Gangolli, the term "Levy-Schoenberg" kernel is defined (Definition 2.3). Consider the ...
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### What is known about "almost orthogonal vectors"?

Motivation: Suppose we have a kernel $k(a,b)$ defined over the natural numbers. Then by the Moore–Aronszajn theorem, we can embedd the natural number $a$ in some Hilbert space $\mathbb{H}$, which we ...
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### abc conjecture and surjective polynomials

Let $a,b$ be coprime multivariate polynomials with integer coefficients and $\deg(a) > \deg(\rm{rad}(a b))$. Let $c=a+b$ and assume $c$ is either surjective or $c$ represents infinitely many ...
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### The abc conjecture modulo variety

It is known that the abc conjecture can't fail with polynomial identities. Is the following special case of abc known? Let $a,b,c,f$ be polynomials with integer coefficients satisfying $a+b=c+f$. ...
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### Anabelian geometry ~ higher category theory

Note: I'm worried this question might be taken as controversial, because it relates to Shinichi Mochizuki’s work on the abc conjecture. However, my question has nothing to do with the correctness of ...
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