Skip to main content

Questions tagged [abc-conjecture]

For questions about and related to the abc conjecture.

Filter by
Sorted by
Tagged with
2 votes
0 answers
66 views

Approximating the logarithms of primes elegantly

What's the "most efficient" way to encode that $\ln(2):\ln(3):\ln(5):\ln(7) \approx 171:271:397:480$ using $3$ approximations? For example: $((\frac{3^3}{5^2})^3)^3 \approx 2$ uses $3+2+3+3 ...
Chinmay The Math Guy's user avatar
2 votes
0 answers
132 views

Spectrum of "infinite-Gram matrix"?

This question comes out of my interest for positive definite kernels over the natural numbers. (I have collected some kernels with proofs). First let me point to a connection between self-adjoint ...
mathoverflowUser's user avatar
29 votes
5 answers
4k views

How is the logarithm of an integer analogous to the degree of a polynomial?

I've recently been reading Serge Lang's Math Talks for Undergraduates, specifically a section about the abc conjecture. Lang starts by stating and proving the Mason-Stothers Theorem: Let $f,g \in \...
Adam Nelson's user avatar
4 votes
0 answers
265 views

Why should we believe the abc conjecture?

To fix notation and check that my definitions are correct I will first state: abc conjecture: Let $a,b\in\mathbb{N}$ be coprime, $c:=a+b$ , and define the quality of the triple $(a,b,c)$ to equal $q(...
user761852's user avatar
0 votes
1 answer
144 views

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 ...
Yep's user avatar
  • 11
0 votes
0 answers
136 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}}+\...
Pythagorus's user avatar
2 votes
0 answers
134 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 $\...
user759001's user avatar
2 votes
1 answer
121 views

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{...
Odair Creazzo Junior's user avatar
0 votes
1 answer
75 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 ...
MasM's user avatar
  • 641
2 votes
1 answer
193 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 ...
user759001's user avatar
2 votes
0 answers
56 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 ...
Rolandb's user avatar
  • 435
1 vote
1 answer
162 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=...
Pythagorus's user avatar
8 votes
0 answers
207 views

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, ...
user avatar
3 votes
0 answers
243 views

Known attempts to prove the $abc$-conjecture

I have been interested in the $abc$-conjecture recently, because of its marvelous applications to number theory (e.g., Fermat's last theorem, Mordell-Faltings theorem, etc.). I've heard of that there ...
glimpser's user avatar
  • 1,202
0 votes
1 answer
168 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 ...
Pythagorus's user avatar
4 votes
0 answers
122 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 $n$,...
Rolandb's user avatar
  • 435
2 votes
0 answers
406 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 ...
Hans-Peter Stricker's user avatar
3 votes
1 answer
201 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 ...
IntegrateThis's user avatar
1 vote
0 answers
47 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 ...
Jens Wagemaker's user avatar
0 votes
1 answer
239 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}...
whytheheckarewedoingmath's user avatar
15 votes
1 answer
4k 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 ...
PJTraill's user avatar
  • 859
0 votes
0 answers
935 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 ...
Aurora Borealis's user avatar
1 vote
0 answers
232 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 ...
Whizkid95's user avatar
  • 741
1 vote
0 answers
105 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 ...
Jens Wagemaker's user avatar
1 vote
2 answers
166 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 = \...
Ed Pegg's user avatar
  • 21.2k
2 votes
3 answers
877 views

Example of a power of 3 which is close to a power of 2 (Related to music theory and Superparticular ratios)

I'm looking for a power of 3 close to a power of 2. Let's say, what is $(n,m)$ such that $$\left|\frac{2^n}{3^m}-1\right| = \min\left \{\left|\frac{2^i}{3^j}-1 \right|, 1\leq i,j\leq 20\right\} \quad ...
Colas's user avatar
  • 269
0 votes
0 answers
45 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 ...
Sylvain Julien's user avatar
8 votes
4 answers
492 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, ...
user avatar
0 votes
1 answer
110 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,...
Cố Gắng Lên's user avatar
1 vote
0 answers
124 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 ...
Flavian's user avatar
  • 137
1 vote
1 answer
161 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?
Lufamily's user avatar
1 vote
0 answers
71 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 ...
Lehs's user avatar
  • 13.9k
3 votes
1 answer
226 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: ...
Lehs's user avatar
  • 13.9k
2 votes
0 answers
563 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 ...
proofromthebook's user avatar
3 votes
1 answer
248 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 ...
Lehs's user avatar
  • 13.9k
1 vote
1 answer
327 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 ...
Lehs's user avatar
  • 13.9k
1 vote
2 answers
156 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 ...
Oss Ickle's user avatar
1 vote
1 answer
140 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*...
Joe's user avatar
  • 55
0 votes
1 answer
251 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)^...
JMP's user avatar
  • 21.8k