Questions tagged [abc-conjecture]
The abc-conjecture tag has no usage guidance.
78
questions
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abc-conjecture and positive definite kernels, again?
One formulation of the abc-conjecture is:
$$\forall a,b \in \mathbb{N}: \frac{a+b}{\gcd(a,b)}< \operatorname{rad}\left ( \frac{ab(a+b)}{\gcd(a,b)^3}\right )^2 $$
Let us define:
$$K(a,b) := \frac{2(...
1
vote
0
answers
103
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On a subset of the $abc$ triples
The $abc$ conjecture states that, for every positive real $\varepsilon$, there exist only finitely many triples $(a, b, c)$ of coprime positive integers such that $a + b = c$ and
$$c > \...
49
votes
4
answers
22k
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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 ...
2
votes
0
answers
383
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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
$x^3 - 1 = (x - k) y^3$,...
2
votes
0
answers
255
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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
$$
|u\cdot ...
7
votes
0
answers
437
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Is it unconditionally known that abc conjecture can't fail on a variety?
Background: this question gives the identity:
$$(x+z)^5+(y-z)^5 = (-3 x + 4 y)^2 (x + y)^3 + (x+y) f(x,y,z)$$
The curve $C : f(x,y,z)=0$ is genus 1, have infinitely many rational and integral points ...
0
votes
0
answers
265
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Does $abc$ preclude very smooth solutions?
Recall the $abc$-conjecture, which asserts that for any $\epsilon > 0$ there exists a positive number $C(\epsilon)$ such that for any coprime integers $a,b,c$ with $a + b = c$ and $\max\{|a|, |b|, |...
4
votes
1
answer
232
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Ruling out an extremely specific class of Wieferich-like primes
Recall that a prime $p$ is a Wieferich prime if $p^2|2^{p-1}-1$. The only known Wieferich primes are $p=1093$ and $p=3511$. A prime $p$ is a generalized Wieferich prime to base $q$ if $p^2|q^{p-1}-1$.
...
8
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0
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448
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The average power of an integer and a strengthened Fermat Last Theorem
It is well known that the ABC conjecture gives an immediate proof of Fermat Last Theorem (FLT). It seems that it proves something stronger involving not necessarily perfect power, which may still be ...
11
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1
answer
387
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Is this set dense in [0,+∞)?
We define $A=\{ \frac{c}{rad(abc)}: a, b > 0, c=a+b, gcd(a, b)=1 \}$.
Is the set $A$ dense in $[0, +\infty)$?
Does $\overline{A}$ have interior? Here $\overline{A}$ is the closure of $A$.
A well-...
-1
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1
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509
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Questions about the abc conjecture [closed]
Question.
Is there an integer $n_0 \geq 2$ such that $$\left\{\frac{c}{rad(abc)^{n_0}}: a, b >0,\; c=a+b,\; \gcd(a, b)=1\right \}$$ is bounded?
The abc conjecture can directly deduce this ...
1
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0
answers
170
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Asymptotic of rad(abc) in the abc conjecture
The abc conjecture famously predicts that, given any $\epsilon>0$, for all but finitely many positive coprime integers $a,b,c$ with $a+b=c$, the radical $rad(abc)$ (i.e., the product of all prime ...
10
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0
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172
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Is almost every number the sum of two numbers with small radicals?
Define a set of numbers with small radicals (A341645 in OEIS) by
$$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
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1
answer
248
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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
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0
answers
124
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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
134
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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
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174
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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
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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
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1
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157
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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 ...
10
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1
answer
2k
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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}.$$
Define the functions $h(P)$ by $h(1)=1$ and $h(P)=\min(a_1, a_2,\ldots,...
0
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0
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175
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Status of the $n$ conjecture and, as secondary question or reference request, what about a transfer method for this conjecture $n>3$
The n conjecture is a generalization of the abc conjecture. What is the current status of the $n$ conjecture? See also [1]
Question 1. Can you tell us what about the current status of the $n$ ...
1
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1
answer
157
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abc triples with a symmetric condition
Recently, I have asked a question about the balance of abc triples. Since then I have come up with a different idea of a new criterion that somewhat combines balance and magnitude and has two ...
0
votes
1
answer
688
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A soft question on the ABC conjecture
In Nature Vol 580, in an article about Shinichi Mochizuki's proposed proof of the abc-conjecture, there is a formulation saying:
The conjecture roughly states that if a lot of small primes divide ...
2
votes
1
answer
388
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On weaker forms of the abc conjecture from the theory of Hölder and logarithmic means
In this post (the content of this post is now cross-posted from Mathematics Stack Exchange see below) we denote the radical of an integer $n>1$ as the product of disctinct primes dividing it $$\...
3
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0
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248
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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 ...
3
votes
1
answer
308
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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 ...
2
votes
1
answer
355
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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 ...
6
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1
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328
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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 ...
7
votes
0
answers
340
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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 ...
25
votes
1
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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:
$$ \frac{a+b}{\gcd(a,b)}\,\ <\,\ K_{\epsilon}\cdot \text{rad}\...
10
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0
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Two questions around the $abc$-conjecture
Let $d(a,b) = 1-\frac{2 \gcd(a,b)}{a+b}$, $d_{ABC}(a,b) = 1-\frac{2\gcd(a,b)^3}{ab(a+b)}$ be two metrics on natural numbers.
The abc-conjecture can be formulated using these two metrics as:
For ...
0
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0
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260
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On variants of the abc conjecture in terms of Lehmer means
In this post we denote the Lehmer mean of a tuple $\text{x}$ of positive real numbers as $$L_p(\text{x})={\sum_{k=1}^nx_k^p\over\sum_{k=1}^nx_k^{p-1}},$$
see the reference Wikipedia Lehmer mean.
The ...
3
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0
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Are there any references in the literature relating to work on finding a Diophantine equation representing abc
The Davis-Putnam-Robinson-Matiyasevich theorem is:
Diophantine is equivalent to listable
This result has some known applications:
(1) Prime-producing polynomials.
(2) Diophantine statement of the ...
1
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0
answers
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Small radical of $F(g,h)$
Related to this question.
Basically this question asks if the original @Granville proposition
always fails.
Is it true that for all $g,h \in \mathbb{Z}[x]$ s.t. $g,h$ are coprime
and $\deg(\mathrm{...
8
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0
answers
575
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Are there at least one of $a$, $b$, $c$, $a+b$, $b+c$, $c+a$ have h(P) $\le 3$? (was checked up to $10^{18}$)
Given a positive integer $P>1$, let its prime factorization be written $$P=p_1^{a_1}p_2^{a_2}p_3^{a_3}\cdots p_k^{a_k}$$
Define the functions $h(P)$ by $h(1)=1$ and $h(P)=\min(a_1, a_2,\ldots,a_k)$
...
24
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4
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A reinterpretation of the $abc$ - conjecture in terms of metric spaces?
I hope it is appropriate to ask this question here:
One formulation of the abc-conjecture is
$$ c < \text{rad}(abc)^2$$
where $\gcd(a,b)=1$ and $c=a+b$. This is equivalent to ($a,b$ being ...
1
vote
0
answers
185
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How small can $u$ be in the Pell equation $u^2-k^3 v^2=\pm 1$?
Let $k$ be positive integer, not a square and let $u_k,v_k$ be non-trivial
solutions to the Pell equation $u_k^2-k^3 v_k^2=\pm 1$.
Q1 How small $u_k$ can be infinitely often as function $k$?
This ...
3
votes
0
answers
86
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Are there infinitely many primes $p$, positive integers $ k, n $ such that $1 \le n < p$ and $p^k > n.rad(p^{k+1}−n)$?
Among $168$ prime numbers in range $1$ to $10^3$, there are $84$ prime numbers $n$ such that: $p^k> n.rad(p^{k+1}−n)$ where $1 \le n<p$ and $k=2,3,4$. There are also $84$ prime numbers $n$ such ...
4
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1
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Large radical of an integer and three AB conjectures
In this Note, We propose a new definition called "large radical of an integer". Using this definition, three very useful $AB$ conjecture are given.
1. Large counter examples of the ABC conjecture
...
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Is the conjecture $min(A,B) \le rad(ABC)$ new and correct? [closed]
$\DeclareMathOperator\rad{rad}$Conjecture: If $A, B, C$ are positive integers with $\gcd(A, B)=1$, $\gcd(B, C)=1$, and $\gcd(C, A)=1$, and if $A+B=C$, then $\min(A,B) \le \rad(ABC)$.
If the ...
1
vote
1
answer
493
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A generalization of Lander, Parkin, and Selfridge conjecture
My question: Are the conjectures as follows correct?
Given a positive integer $P>1$, let its prime factorization be written
$$P=p_1^{a_1}p_2^{a_2}p_3^{a_3}...p_k^{a_k}$$.
Define the ...
1
vote
0
answers
144
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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 ...
2
votes
0
answers
228
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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
$a+b=c+f$. ...
9
votes
1
answer
2k
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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 ...
10
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0
answers
453
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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
561
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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
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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
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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
453
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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
421
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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 \...