Questions tagged [abc-conjecture]

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40 votes
2 answers
7k views

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 ...
2 votes
0 answers
356 views

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 $x^3 - 1 = (x - k) y^3$,...
2 votes
0 answers
230 views

Baby $abc$ conjecture for $n$-th roots

Is there any progress on a “baby $abc$ conjecture” where you restrict attention to rational approximations of $n$-th roots? Let $r/s$ be a very close approximation to $(t/u)^{1/n}$, so that $$ |u\cdot ...
7 votes
0 answers
420 views

Is it unconditionally known that abc conjecture can't fail on a variety?

Background: this question gives the identity: $$(x+z)^5+(y-z)^5 = (-3 x + 4 y)^2 (x + y)^3 + (x+y) f(x,y,z)$$ The curve $C : f(x,y,z)=0$ is genus 1, have infinitely many rational and integral points ...
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0 votes
0 answers
256 views

Does $abc$ preclude very smooth solutions?

Recall the $abc$-conjecture, which asserts that for any $\epsilon > 0$ there exists a positive number $C(\epsilon)$ such that for any coprime integers $a,b,c$ with $a + b = c$ and $\max\{|a|, |b|, |...
4 votes
1 answer
220 views

Ruling out an extremely specific class of Wieferich-like primes

Recall that a prime $p$ is a Wieferich prime if $p^2|2^{p-1}-1$. The only known Wieferich primes are $p=1093$ and $p=3511$. A prime $p$ is a generalized Wieferich prime to base $q$ if $p^2|q^{p-1}-1$. ...
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8 votes
0 answers
386 views

The average power of an integer and a strengthened Fermat Last Theorem

It is well known that the ABC conjecture gives an immediate proof of Fermat Last Theorem (FLT). It seems that it proves something stronger involving not necessarily perfect power, which may still be ...
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11 votes
1 answer
377 views

Is this set dense in [0,+∞)?

We define $A=\{ \frac{c}{rad(abc)}: a, b > 0, c=a+b, gcd(a, b)=1 \}$. Is the set $A$ dense in $[0, +\infty)$? Does $\overline{A}$ have interior? Here $\overline{A}$ is the closure of $A$. A well-...
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-1 votes
1 answer
467 views

Questions about the abc conjecture [closed]

Question. Is there an integer $n_0 \geq 2$ such that $$\left\{\frac{c}{rad(abc)^{n_0}}: a, b >0,\; c=a+b,\; \gcd(a, b)=1\right \}$$ is bounded? The abc conjecture can directly deduce this ...
  • 517
1 vote
0 answers
137 views

Asymptotic of rad(abc) in the abc conjecture

The abc conjecture famously predicts that, given any $\epsilon>0$, for all but finitely many positive coprime integers $a,b,c$ with $a+b=c$, the radical $rad(abc)$ (i.e., the product of all prime ...
9 votes
0 answers
158 views

Is almost every number the sum of two numbers with small radicals?

Define a set of numbers with small radicals (A341645 in OEIS) by $$A_2=\{n\in\mathbb{N} \;|\; \text{rad}(n)^2\le n\}$$ The asymptotic density of $A_2\cap \{1,\dots N\}$ is $\sqrt{N}\times e^{2(1+o(1))\...
0 votes
0 answers
185 views

The d(abc)-theorem, the abc-conjecture and positive definite kernels over the natural numbers?

I noticed in the work of Hector Pasten, Th.1.11 the $d(abc)$ theorem and have a question, after doing some experiments with sagemath. Let $s_k(n) = \sum_{d|n}{ d^k }$ be the sum of divisiors $d$ of $n$...
0 votes
0 answers
116 views

How small the radical of $xyz(x+y+z)$ can be infinitely?

This is an open problem. Let $x,y,z$ be coprime integers (not necessarily pairwise coprime) and no proper subset sum of $\{x,y,z,-(x+y+z)\}$ is zero. For a quadruple $(x,y,z,-(x+y+z))$ define the ...
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0 votes
0 answers
103 views

A conjectural limit involving primorial and factorial

It is well known that the abc conjecture implies that the there are only finitely many solutions to Brocard problem, as shown by Overholt in Overholt, Marius (1993), "The diophantine equation $n! ...
1 vote
2 answers
171 views

How small the radical of $xy(x+y)uv(u+v)$ can be infinitely often?

Let $x,y,u,v$ be positive integers with $x,y$ coprime and $u,v$ coprime ( $xy,uv$ not necessarily coprime). Assume $x+y \ne u+v$. How small the radical of $xy(x+y)uv(u+v)$ can be infinitely often? Can ...
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2 votes
1 answer
3k views

Berkeley mathematics department colloquium by S.Mochizuki [closed]

On the website of the Berkeley mathematics department there is mention (see this) of a colloquium held on november 5, 2020 (by Zoom) whose speaker was Shinichi Mochizuki, with a talk titled "...
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2 votes
1 answer
138 views

a b c triples with bounded prime factors

(i) For any fixed $B>0$, are there only finitely many triples $a,b,c$ of coprime positive integers, such that $a+b=c$ and all prime factors of $a,b,c$ are at most $B$? (ii) For which $B$ all such ...
  • 713
9 votes
1 answer
1k views

Is new $n$-conjecture as follows correct?

Given a positive integer $P>1$, let its prime factorization be written as$$P=p_1^{a_1}p_2^{a_2}p_3^{a_3}\cdots p_k^{a_k}.$$ Define the functions $h(P)$ by $h(1)=1$ and $h(P)=\min(a_1, a_2,\ldots,...
0 votes
0 answers
173 views

Status of the $n$ conjecture and, as secondary question or reference request, what about a transfer method for this conjecture $n>3$

The n conjecture is a generalization of the abc conjecture. What is the current status of the $n$ conjecture? See also [1] Question 1. Can you tell us what about the current status of the $n$ ...
1 vote
1 answer
148 views

abc triples with a symmetric condition

Recently, I have asked a question about the balance of abc triples. Since then I have come up with a different idea of a new criterion that somewhat combines balance and magnitude and has two ...
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0 votes
1 answer
642 views

A soft question on the ABC conjecture

In Nature Vol 580, in an article about Shinichi Mochizuki's proposed proof of the abc-conjecture, there is a formulation saying: The conjecture roughly states that if a lot of small primes divide ...
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2 votes
1 answer
329 views

On weaker forms of the abc conjecture from the theory of Hölder and logarithmic means

In this post (the content of this post is now cross-posted from Mathematics Stack Exchange see below) we denote the radical of an integer $n>1$ as the product of disctinct primes dividing it $$\...
3 votes
0 answers
235 views

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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3 votes
1 answer
284 views

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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2 votes
1 answer
293 views

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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6 votes
1 answer
297 views

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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7 votes
0 answers
247 views

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 ...
user avatar
21 votes
1 answer
1k views

The abc-conjecture as an inequality for inner-products?

The abc-conjecture is: For every $\epsilon > 0$ there exists $K_{\epsilon}$ such that for all natural numbers $a \neq b$ we have: $$ \frac{a+b}{\gcd(a,b)}\,\ <\,\ K_{\epsilon}\cdot \text{rad}\...
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10 votes
0 answers
762 views

Two questions around the $abc$-conjecture

Let $d(a,b) = 1-\frac{2 \gcd(a,b)}{a+b}$, $d_{ABC}(a,b) = 1-\frac{2\gcd(a,b)^3}{ab(a+b)}$ be two metrics on natural numbers. The abc-conjecture can be formulated using these two metrics as: For ...
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1 vote
0 answers
240 views

On variants of the abc conjecture in terms of Lehmer means

In this post we denote the Lehmer mean of a tuple $\text{x}$ of positive real numbers as $$L_p(\text{x})={\sum_{k=1}^nx_k^p\over\sum_{k=1}^nx_k^{p-1}},$$ see the reference Wikipedia Lehmer mean. The ...
3 votes
0 answers
260 views

Are there any references in the literature relating to work on finding a Diophantine equation representing abc

The Davis-Putnam-Robinson-Matiyasevich theorem is: Diophantine is equivalent to listable This result has some known applications: (1) Prime-producing polynomials. (2) Diophantine statement of the ...
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1 vote
0 answers
39 views

Small radical of $F(g,h)$

Related to this question. Basically this question asks if the original @Granville proposition always fails. Is it true that for all $g,h \in \mathbb{Z}[x]$ s.t. $g,h$ are coprime and $\deg(\mathrm{...
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8 votes
0 answers
571 views

Are there at least one of $a$, $b$, $c$, $a+b$, $b+c$, $c+a$ have h(P) $\le 3$? (was checked up to $10^{18}$)

Given a positive integer $P>1$, let its prime factorization be written $$P=p_1^{a_1}p_2^{a_2}p_3^{a_3}\cdots p_k^{a_k}$$ Define the functions $h(P)$ by $h(1)=1$ and $h(P)=\min(a_1, a_2,\ldots,a_k)$ ...
24 votes
4 answers
2k views

A reinterpretation of the $abc$ - conjecture in terms of metric spaces?

I hope it is appropriate to ask this question here: One formulation of the abc-conjecture is $$ c < \text{rad}(abc)^2$$ where $\gcd(a,b)=1$ and $c=a+b$. This is equivalent to ($a,b$ being ...
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1 vote
0 answers
180 views

How small can $u$ be in the Pell equation $u^2-k^3 v^2=\pm 1$?

Let $k$ be positive integer, not a square and let $u_k,v_k$ be non-trivial solutions to the Pell equation $u_k^2-k^3 v_k^2=\pm 1$. Q1 How small $u_k$ can be infinitely often as function $k$? This ...
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3 votes
0 answers
82 views

Are there infinitely many primes $p$, positive integers $ k, n $ such that $1 \le n < p$ and $p^k > n.rad(p^{k+1}−n)$?

Among $168$ prime numbers in range $1$ to $10^3$, there are $84$ prime numbers $n$ such that: $p^k> n.rad(p^{k+1}−n)$ where $1 \le n<p$ and $k=2,3,4$. There are also $84$ prime numbers $n$ such ...
4 votes
1 answer
477 views

Large radical of an integer and three AB conjectures

In this Note, We propose a new definition called "large radical of an integer". Using this definition, three very useful $AB$ conjecture are given. 1. Large counter examples of the ABC conjecture ...
-2 votes
1 answer
244 views

Is the conjecture $min(A,B) \le rad(ABC)$ new and correct? [closed]

$\DeclareMathOperator\rad{rad}$Conjecture: If $A, B, C$ are positive integers with $\gcd(A, B)=1$, $\gcd(B, C)=1$, and $\gcd(C, A)=1$, and if $A+B=C$, then $\min(A,B) \le \rad(ABC)$. If the ...
1 vote
1 answer
433 views

A generalization of Lander, Parkin, and Selfridge conjecture

My question: Are the conjectures as follows correct? Given a positive integer $P>1$, let its prime factorization be written $$P=p_1^{a_1}p_2^{a_2}p_3^{a_3}...p_k^{a_k}$$. Define the ...
1 vote
0 answers
139 views

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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2 votes
0 answers
217 views

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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9 votes
1 answer
2k views

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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10 votes
0 answers
440 views

Mini-$abc$ conjecture

Define $\text{rad}_{23}(2^m3^nr)=2^{\text{sign}(m)}3^{\text{sign}(n)}r$, where $m,n\ge0$ and $2,3\nmid r\in\mathbb{N}$. For a triple $a+b=c$ define the quality $q_{23}(a,b,c)=\frac{\log(c)}{\log(\...
2 votes
0 answers
556 views

Can the ABC conjecture be expanded?

Has anyone considered expanding the range of terms $a$ and $b$ for each $c$? I have generated triples $(a, b, c)$ that form integer triangles including the degenerate case of $a + b = c$ such that $a ...
9 votes
0 answers
4k views

Is the conjecture A+B=C following correct?

Is the conjecture on A+B=C following correct ? Conjecture: Let $A, B, C$ be three positive integer numbers such that $A+B=C$ with $\gcd(A, B, C) = 1$. By Fundamental theorem of arithmetic we write: $...
14 votes
0 answers
1k views

Explicit example of elliptic curve of the kind needed for IUTT

At the nLab, we are currently trying to illustrate the definition of initial theta-data in Mochizuki's first IUTT paper by means of an explicit example. The exposition should end up at the following ...
4 votes
2 answers
432 views

Small $|2^x 3^y - 5^z 7^t|$ and generalization

Let $\{p_i\},\{q_i\}$ be disjoint sets of primes. For natural $e_i,f_i$ define $A=\prod p_i^{e_i},B=\prod q_i^{f_i}$. Is it true that for all real $d < 1$, $|A-B| < \max(A,B)^d$ has finitely ...
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6 votes
1 answer
412 views

Strengthening an implication of the abc conjecture

Granville gives p.5 an implication of the abc conjecture: Assume the abc conjecture. Let $f(x,y)$ be squarefree homogeneous polynomial with integer coefficients. For coprime integers $m,n$ if $q^2 \...
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24 votes
1 answer
1k views

Another weak form of the ABC conjecture

First I will explain why a weaker form is needed. And then I formulate the conjecture (more precisely, the formulation will be clear). It is related to the question https://math.stackexchange.com/...
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1 vote
1 answer
232 views

abc streams (sequences of creek stones)

A sequence of natural numbers $\ (c_n: n=1\ 2\ \ldots)\ $ is called a sequence of creek stones $\ \Leftarrow:\Rightarrow\ \forall_{n=1\ 2\ \ldots}\,c_{n+1}\ge c_n^2\ $. Given natural $\ a\ b,\ $ ...