# Questions tagged [abc-conjecture]

The abc-conjecture tag has no usage guidance.

55
questions

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51 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 ...

**5**

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**0**answers

283 views

### Is new n-conjecture as follows correct?

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

**0**

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158 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 $n$ ...

**2**

votes

**1**answer

130 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 ...

**-1**

votes

**1**answer

474 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

244 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 $$\...

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192 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

244 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

195 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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**1**answer

267 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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213 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 ...

**20**

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**

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735 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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217 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 ...

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242 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**

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**0**answers

38 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**

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**0**answers

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

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**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**

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**0**answers

173 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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77 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 ...

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votes

**1**answer

448 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
...

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**1**answer

223 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

356 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**

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126 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**

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**0**answers

205 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**

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**0**answers

544 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 ...

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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:
$...

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**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**

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**2**answers

399 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

396 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

226 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,\ $ ...

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**0**answers

760 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 ...

**4**

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'...

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**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**

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**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 ...

**63**

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**6**answers

22k 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**

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

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**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-...

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356 views

### The ABC conjecture where A and B are smooth

Mochizuki has already claimed to have proven the ABC-conjecture. But even if his claim turns out to be correct the proof will not be easy to understand. With that in mind I'm asking wether anything is ...

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155 views

### Radical of a polynomial values

It has been observed by Langevin and Elkies that the following holds:
Assume that the $ABC$- conjecture is true. Suppose that $f(x)\in\mathbb{Z}[x]$ has no repeated roots. Fix $\varepsilon >0.$ ...

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votes

**1**answer

829 views

### Application of the Riemann hypothesis and the ABC conjecture to independence results

In Old Home Page of Andreas Weiermann Andreas Weiermann has stated the following:
Quite recently I submitted a preprint about an application of the Riemann hypothesis and the ABC conjecture to ...

**4**

votes

**2**answers

563 views

### Argument againts the $abcd$ conjecture with extra gcd conditions

Got an argument and numerical support againts the $abcd$
conjecture with extra gcd conditions (observe that this
is different from the $abc$ and the $abcd$ conjectures).
This thesis p. 20 defines the ...

**1**

vote

**0**answers

214 views

### Is it trivial to get $200$ algebraic abc triples of equal quality $1.6978…$ over isomorphic number fields?

Got $200$ algebraic abc triples over distinct though isomorphic
number fields of equal quality $1.6978...$
Strongly suspect I can get as many as I like
(assuming the computations are correct).
Is ...

**17**

votes

**3**answers

953 views

### 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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**0**answers

365 views

### OPN, Nagell-Ljunggren equation and ABC conjecture

Following my answer to Algebraic Attacks on the Odd Perfect Number Problem, I would like to know whether the argument of quid, namely that if a hypothetic odd perfect number $n$ is such that $\...

**32**

votes

**2**answers

1k views

### 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(\...

**2**

votes

**0**answers

192 views

### Is there a generalization of Granville-Langevin conjecture for number fields?

According to Wikipedia and other sources the Granville-Langevin conjecture
states:
If $f$ is a square-free binary form of degree $n > 2$, then for every real $\beta > 2$ there is a constant $...

**9**

votes

**3**answers

863 views

### Reference request for $(1,2^n-1,2^n)$ example related to abc-conjecture

The $abc$-conjecture states that if $a,b,c$ are positive, relatively prime integers satisfying $a+b=c$, then the product of the primes dividing $abc$ (the radical of $abc$) is $\gg_\varepsilon c^{1-\...