Questions tagged [abc-conjecture]

For questions about and related to the abc conjecture.

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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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0answers
66 views

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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0answers
126 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
138 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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0answers
33 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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1answer
84 views

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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1answer
2k views

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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0answers
548 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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0answers
77 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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81 views

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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2answers
114 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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1answer
785 views

Is this a spam message or the Riemann hypothesis really has been proved? [closed]

I have received now message from so@one-zero.eu include paper show that the Auther proved the Riemann hypothesis as shown here in this paperwhich include both relation between the distribution of the ...
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0answers
39 views

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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4answers
404 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
101 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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0answers
98 views

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
108 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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59 views

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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1answer
193 views

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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0answers
378 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
200 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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1answer
235 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
135 views

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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1answer
93 views

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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1answer
152 views

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)^...