Questions tagged [abc-conjecture]

For questions about and related to the abc conjecture.

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Serge Lang Algebra - Generalized Szpiro Conjecture-like inequality derived from ABC Conjecture

There is a specific part of the section of Serge Lang's Algebra from Chapter IV Section 7 regarding an inequality preceding the generalized Szpiro Conjecture that confused me. How exactly are the ...
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82 views

$ABC$ Conjecture: Stewart & Yu 2001 result. Is Wikipedia mistaken?

In the Wikipedia article on the $ABC$ conjecture in the theoretical results section, they give the Stewart & Yu 2001 result as: $$c<\exp \left(K_{3}\operatorname {rad} (abc)^{{\frac {1}{3}}+\...
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107 views

A weak form of the abc conjecture involving the definition of Hölder mean

I wondered about a weak form of the abc conjecture, see the Wikipedia abc conjecture using the theory of generalized means, I mean this Wikipedia Generalized mean. We get the following claim, where $\...
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73 views

How do you get the $6/5$ in Baker's explicit $abc$ conjecture?

I can't find Baker's "Experiments on the ABC conjecture" in which he gives the $6/5$ as the absolute constant. How do you get the $6/5$ as the constant? Baker's conjecture: For $a+b=c$, $(a,b,c)=1$...
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1answer
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ABC-conjecture: is the 3rd definition in Wikipedia really valid?

From Wikipedia for 'abc-conjecture': "A third equivalent formulation of the conjecture involves the quality $q(a, b, c)$ of the triple $(a, b, c)$, defined as $q(a,b,c)= \frac{\log(c)}{\log(\text{...
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1answer
59 views

does a slight change in a real value cause a massive change from finite to infinite?

Can anybody give an example of a system of equivalence and/or inequality relations dependent to just a real value number like $\kappa$, which satisfies for finitely many integers if $\kappa =1$ and ...
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49 views

For an exceptional quality $q$, can $q-1$ be part of an infinite family of qualities of $ABC$ triples?

Consider the $ABC$ conjecture in the form: For every $\epsilon>0$ $q>1+\epsilon$ in a finite number of triples $(a,b,c)$ where $a+b=c$ $gcd(a,b,c)=1$ and $q=log(c)/log(radical(abc)).$ An ...
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152 views

Weaker than abc conjecture invoking the inequality between the arithmetic and logarithmic means

In this post we denote the radical of an integer $n>1$ as $$\operatorname{rad}(n)=\prod_{\substack{p\mid n\\p\text{ prime}}}p$$ with the definition $\operatorname{rad}(1)=1$. The abc conjecture is ...
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48 views

Can we prove that $\operatorname{rad}(ABC)^2-C$ is not a square when $A+B=C$ are coprime integers?

A weak form of the ABC conjecture states that $\operatorname{rad}(ABC)^2-C$ is always positive. Here the radical of a positive integer $n$, denoted $\operatorname{rad}(n)$, is the product of the ...
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1answer
102 views

Is the constant in the abc-conjecture dependent on $\epsilon$ only?

Consider the ABC conjecture in the following form: For every positive real number $ϵ$, there exists a constant $k_\epsilon$ such that for all triples $(a,b,c)$ of coprime positive integers, with $a+b=...
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A generalization of the abc-conjecture?

For a natural number $a$ define $X_a := \{a/k| 1\le k \le a \}$. Then it is not difficult to show that $|X_a \cap X_b| = \gcd(a,b)$. Using this one can define similarities over the natural numbers, ...
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131 views

Is this “metric” matrix positive semidefinite?

For a natural number $a$ define $X_a := \{a/k| 1\le k \le a \}$. Then it is not difficult to show that $|X_a \cap X_b| = \gcd(a,b)$. Using this one can define similarities over the natural numbers, ...
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142 views

Known attempts to prove the abc conjecture

I have interested in the abc conjecture recently, because of its marvelous application to number theory.(e.g., Fermat's last theorem, Mordell-Faltings theorem, etc.) I've heard of that there is a ...
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1answer
138 views

What is the implication of a particular choice of $k(\epsilon)$ for the ABC Conjecture.

Consider the $ABC$ conjecture in the following form: For every positive real number $\epsilon$, there exists a constant $k(\epsilon)$ such that for all triples $(a, b, c)$ of coprime positive ...
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Can we prove that for ABC-triples the product $A*B*C$ is unique?

Consider positive coprime integers $A$ and $B$ with $A+B=C$. The triple $(A,B,C)$ is called an ABC-triple if the radical of the product $ABC$ is smaller then $C$. The radical of a positive integer $n$,...
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224 views

The $abc$ conjecture as a special case of Vojta's height inequality

From Quanta I've learned that Peter Scholze and Jakob Stix rejected Shinichi Mochizuki's proof of the $abc$ conjecture in September 2018. As a non-expert one stumbles a little earlier than necessary ...
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1answer
155 views

ABC conjecture and an inequality

Problem: Let $p,q,r$, be positive integers satisfying $\frac {1}{p} + \frac {1}{q} + \frac {1}{r} < 1$ . If the ABC conjecture is true, then $x^p + y^q = z^r$ has finitely many positive integer ...
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37 views

given $p,q,r \ge 3$ study the diophantine equation $x^py^q=z^r-1$ using the $abc$-conjecture

I want to show that given $p,q,r \ge 3$ the diophantine equation $x^py^q=z^r-1$ has only finitely many solutions with $x,y,z \in \mathbb{N} = 1 ,2, \dots$ assuming the $abc$-conjecture. The proof ...
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ABC conjecture exceptions

I’m working on a simple presentation for a class on the ABC conjecture. Nothing too deep. I would like to give an example of a triplet of coprime integers $a, b, c$ such that $$c<\operatorname{rad}...
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Why does Mochizuki insist on “forgetting the previous history of an object”?

What is the simplest (at the lowest level feasible) explanation of the approach of “forgetting the history” of a mathematical object, as used in Inter-universal Teichmüller Theory (IUTT)? Please ...
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649 views

ABC conjecture Proof

So I have a question. Why is Shinichi Mochizuki's proof on the conjecture still not accepted? It has been 6 years, surely there must be some approval or disapproval regarding his proof? Can anyone ...
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129 views

Square-free values of polynomials of degree 3

While reading an article by Dan Carmon on Square-free values of large polynomials over the rational function field, I tried to prove that square-free polynomials of degree $1$ and $2$ have a positive ...
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How can the abc conjecture put constraints on the prime factors of $c$?

Let $c = a + b$ be an abc-triple $(a, b, c) \in \mathbb{N}$ with $\text{gcd}(a,b,c) = 1$, with the prime factorizations of $a$ and $b$ known. How can the abc conjecture put constraints on the prime ...
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126 views

Complex number ABC Conjecture

Regarding the abc conjecture, I can't find records for Gaussian integers. Let $rad(a) = \lVert{\prod{prime factors(a)}}\rVert$. For relatively prime $(a,b,c)$ with $a+b=c$, define quality as $Q = \...
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Does it follow from abc conjecture that $\Omega(p^{2}-1)>max(\Omega((p-1)^2-1),\Omega((p+1)^2-1) $?

It seems that when $ n $ is prime, the map $ f $ that maps an integer $ n $ to the number $ \Omega(n^2-1) $ of primes dividing the argument counted with multiplicity is larger than the same ...
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440 views

A conjecture inspired by the abc-conjecture

This conjecture is obviously inspired by the abc-conjecture: Let $\gcd(a,b)=1$ then $\operatorname{rad}((a+b)ab(ab+a+b))> ab+a+b$ I am not asking for a proof, just for possible counterexamples, ...
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1answer
102 views

A conjecture on diophantine equation

Let $A, B, C$ be given three integer numbers, $A^2+B^2+C^2 \ne 0$; and $f(x,y,z)=0$ be given diophantine equation. Then exist $N_0 > 2$ such that $m, n, k \ge N_0$ the equation as follows: $f(x,...
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What mathematical conjectures broke down in an unintuitive way? [duplicate]

Conjectures are often based on inference from a few cases. A famous example is the Riemann Hypothesis which states that all zeros of the zeta function have real part 1/2. All the zeros that we've ...
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1answer
122 views

When we try to solve the hard problem like 'ABC conjecture',should we first examine if it is a proposition that neither prove nor disprove [closed]

In mathematic logic,Godel's incompleteness theorem tell us that a proposition that neither prove nor disprove is exist.so did every famous hard problem need to be examine first?
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A variant of the radical for integers and the abc-conjecture

Define $\mathcal P(n)=\{\{q_1\},\dots,\{q_m\}\}$ where $n=\prod_{i=1}^mq_i$ and all $q_i\in\mathbb P$. Then one can define $\displaystyle \operatorname{rad}(n)=\prod_{p\in\bigcup\mathcal P(n)}p$ and ...
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Conjecture related to abc-conjecture, with $a<b=p^n,\,n>1$

Let $\operatorname{rad}(b)$ be the product of all distinct prime factors of $b$. The numbers $\,a,b,c\,$ is a $abc$-triple if they are coprime and $a+b=c$. One version of the abc-conjecture is then: ...
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440 views

Video of Shinichi Mochizuki skype conference

In late 2015, Shinichi Mochizuki gave a skype conference to mathematicians about his IU Teichmuller theory. I have tried to find videos of that conference but my efforts have been in vain. Do you ...
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1answer
214 views

Conjecture: injection from exceptional abc-triplets to natural numbers

My question A conjecture with connection to the $abc$-conjecture is about a conjectured injection from exceptional $abc$-triplets $(a,b,c)\mapsto a^2+b^2$, but this question is about a ...
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262 views

A conjecture with connection to the $abc$-conjecture

The numbers $\,a,b,c\,$ is a $abc$-triplet if they are coprime and $a+b=c$. One version of the abc-conjecture is then: For all $\varepsilon>0$ the set $E_\varepsilon$ of all $abc$-triplets with ...
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2answers
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Hoping for clarification of this very basic explanation of the abc conjecture

An article online says: The abc conjecture refers to numerical expressions of the type $a + b = c$. The statement, which comes in several slightly different versions, concerns the prime numbers ...
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Prove for $abc$-triples that $c\leq rad(abc)^2$

Prove for $abc$-triples that $c\leq \text{rad}(abc)^2$. $\text{rad}(abc)$ and $\text{rad}(x)$ means here the product of all prime factors of $x$. (edit 2) The above holds for: $16+5=21 \leq (2*5*3*...
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An approach on the $abc$-conjecture

abc-conjecture ABC Conjecture. For every $\varepsilon>0$, there exist only finitely many triples (a, b, c) of positive coprime integers, with a + b = c, such that: $$c\gt\operatorname{rad}(abc)^...