Questions tagged [abc-conjecture]

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4
votes
1answer
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
1answer
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
1answer
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
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0answers
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
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0answers
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
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0answers
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
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0answers
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
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0answers
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
2answers
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
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1answer
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
1answer
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
1answer
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
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0answers
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
1answer
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
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1answer
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
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1answer
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
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0answers
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
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1answer
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
1answer
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
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1answer
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
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0answers
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
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1answer
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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0answers
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
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0answers
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
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0answers
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
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0answers
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
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0answers
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
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4answers
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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0answers
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
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0answers
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
1answer
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
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1answer
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
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1answer
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
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0answers
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
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0answers
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
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1answer
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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0answers
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
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0answers
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
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0answers
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
2answers
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
1answer
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
1answer
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
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0answers
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
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1answer
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
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1answer
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
1answer
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
6answers
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
1answer
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
0answers
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
1answer
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-...