# Questions tagged [abc-conjecture]

The abc-conjecture tag has no usage guidance.

65
questions

**4**

votes

**1**answer

185 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 $p^2|q^{p-1}-1$.
...

**11**

votes

**1**answer

360 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-...

**-1**

votes

**1**answer

424 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 ...

**0**

votes

**0**answers

116 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

144 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

161 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

105 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 ...

**0**

votes

**0**answers

89 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

170 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 ...

**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 "...

**2**

votes

**1**answer

134 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 ...

**8**

votes

**1**answer

998 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

167 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

144 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 ...

**0**

votes

**1**answer

597 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 ...

**3**

votes

**1**answer

266 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

211 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 ...

**3**

votes

**1**answer

268 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 ...

**2**

votes

**1**answer

240 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 ...

**6**

votes

**1**answer

277 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 ...

**7**

votes

**0**answers

224 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 ...

**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}\...

**10**

votes

**0**answers

750 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 ...

**1**

vote

**0**answers

233 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

252 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 ...

**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{...

**8**

votes

**0**answers

561 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 $h(P)=\min(a_1, a_2,\ldots,a_k)$
...

**22**

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 ...

**1**

vote

**0**answers

176 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 ...

**3**

votes

**0**answers

78 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

462 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

237 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 ...

**2**

votes

**1**answer

395 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

131 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 ...

**2**

votes

**0**answers

211 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
$a+b=c+f$. ...

**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 ...

**2**

votes

**0**answers

550 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

412 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 ...

**6**

votes

**1**answer

404 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 \...

**1**

vote

**1**answer

230 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,\ $ ...

**1**

vote

**0**answers

777 views

### Proof of the ABC conjecture - how feasible would it be to automate some of the deciphering of Shinichi Mochizuki’s proof?

This is a question I will come back to. I am very interested in Shinichi Mochizuki’s proof, and in particular, the idiosyncrasies of his notation, which I understand to be at the root of why it is ...

**5**

votes

**1**answer

2k views

### ABC conjecture and Fermat's last theorem

I have frequently read and heard that given the ABC-conjecture a number of important unsolved problems of number theory can be solved (with relatively simple proofs). Among them, the celebrated Fermat'...

**11**

votes

**1**answer

2k views

### Mochizuki's Gaussian Integral Analogy

In his latest 115-page overview, Mochizuki spends some time explaining "alien copies" by the analogue of evaluating the Gaussian integral by squaring it and introducing a second variable/dimension. In ...

**29**

votes

**1**answer

4k views

### What was achieved on IUT summit, RIMS workshop? [closed]

I would like to know what was achieved in the workshop towards the verification of abc conjecture's proof and the advance of understanding of IUT in general.
A comment from a participant:
C ...

**66**

votes

**6**answers

24k views

### Have there been any updates on Mochizuki's proposed proof of the abc conjecture?

In August 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 ...

**17**

votes

**1**answer

1k views

### 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 ...

**17**

votes

**0**answers

1k views

### 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
$$a+b = c = \mathrm{rad}(abc)^{q(a,b,c)}.$$...

**0**

votes

**1**answer

3k views

### Can one expect the existence of a relevant approach for a proof of the Riemann hypothesis using Mochizuki's theory? [closed]

Next month at Oxford university, there will have the first workshop outside Asia on the Inter-Universal Teichmuller theory of Shinichi Mochizuki: http://www.claymath.org/events/iut-theory-shinichi-...