Questions tagged [abc-conjecture]

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4 votes
2 answers
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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 ...
joro's user avatar
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6 votes
1 answer
433 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 \...
joro's user avatar
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24 votes
1 answer
1k views

Another weak form of the ABC conjecture

First I will explain why a weaker form is needed. And then I formulate the conjecture (more precisely, the formulation will be clear). It is related to the question https://math.stackexchange.com/...
user avatar
1 vote
1 answer
282 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,\ $ such ...
Włodzimierz Holsztyński's user avatar
1 vote
0 answers
825 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 ...
Seraphina's user avatar
  • 169
5 votes
1 answer
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'...
Konstantinos Kanakoglou's user avatar
11 votes
1 answer
3k 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 ...
post.as.a.guest's user avatar
29 votes
1 answer
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 ...
70 votes
7 answers
27k 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
1 answer
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 ...
Stefan Kohl's user avatar
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17 votes
0 answers
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 $$a+b = c = \mathrm{rad}(abc)^{q(a,b,c)}.$$...
Sebastien Palcoux's user avatar
0 votes
1 answer
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-...
Sebastien Palcoux's user avatar
5 votes
0 answers
402 views

The ABC conjecture where A and B are smooth

Mochizuki has already claimed to have proven the ABC-conjecture. But even if his claim turns out to be correct the proof will not be easy to understand. With that in mind I'm asking wether anything is ...
Maarten Derickx's user avatar
5 votes
0 answers
174 views

Radical of a polynomial values

It has been observed by Langevin and Elkies that the following holds: Assume that the $ABC$- conjecture is true. Suppose that $f(x)\in\mathbb{Z}[x]$ has no repeated roots. Fix $\varepsilon >0.$ ...
sergey's user avatar
  • 221
9 votes
1 answer
908 views

Application of the Riemann hypothesis and the ABC conjecture to independence results

In Old Home Page of Andreas Weiermann Andreas Weiermann has stated the following: Quite recently I submitted a preprint about an application of the Riemann hypothesis and the ABC conjecture to ...
Mohammad Golshani's user avatar
5 votes
2 answers
695 views

Argument againts the $abcd$ conjecture with extra gcd conditions

Got an argument and numerical support againts the $abcd$ conjecture with extra gcd conditions (observe that this is different from the $abc$ and the $abcd$ conjectures). This thesis p. 20 defines the ...
joro's user avatar
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1 vote
0 answers
225 views

Is it trivial to get $200$ algebraic abc triples of equal quality $1.6978...$ over isomorphic number fields?

Got $200$ algebraic abc triples over distinct though isomorphic number fields of equal quality $1.6978...$ Strongly suspect I can get as many as I like (assuming the computations are correct). Is ...
joro's user avatar
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17 votes
3 answers
1k views

Probing the generalization of the abc conjecture to more than 3 variables

Browkin and Brzezinski, in "Some remarks on the $abc$-conjecture", Math. Comp. 62 (1994), no. 206, 931–939, state the following generalization of the $abc$ conjecture to more than three variables: ...
Greg Martin's user avatar
  • 12.7k
2 votes
0 answers
381 views

OPN, Nagell-Ljunggren equation and ABC conjecture

Following my answer to Algebraic Attacks on the Odd Perfect Number Problem, I would like to know whether the argument of quid, namely that if a hypothetic odd perfect number $n$ is such that $\...
Sylvain JULIEN's user avatar
35 votes
2 answers
2k views

Difference of j-invariant values and the abc conjecture

I learned of the following example in a recent seminar: if $j(\tau)$ denotes the usual $j$-invariant, and $\alpha = (-1+i\sqrt{163})/2$, then \begin{align*} \frac{j(i)}{1728} &= 1 \\ \frac{-j(\...
Greg Martin's user avatar
  • 12.7k
2 votes
0 answers
229 views

Is there a generalization of Granville-Langevin conjecture for number fields?

According to Wikipedia and other sources the Granville-Langevin conjecture states: If $f$ is a square-free binary form of degree $n > 2$, then for every real $\beta > 2$ there is a constant $...
joro's user avatar
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8 votes
3 answers
881 views

Reference request for $(1,2^n-1,2^n)$ example related to abc-conjecture

The $abc$-conjecture states that if $a,b,c$ are positive, relatively prime integers satisfying $a+b=c$, then the product of the primes dividing $abc$ (the radical of $abc$) is $\gg_\varepsilon c^{1-\...
Greg Martin's user avatar
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10 votes
3 answers
1k views

Why is the gcd so large in an identity related to the $abc$ conjecture?

Consider the identity $$ (x+z)^5+(y-z)^5 = (-3 x + 4 y)^2 (x + y)^3 + (x+y) f(x,y,z) $$ Where $f(x,y,z)=(-8*x^4 + 5*x^3*y + 24*x^2*y^2 - 9*x*y^3 - 15*y^4 + 5*x^3*z - 5*x^2*y*z + 5*x*y^2*z - 5*y^3*z + ...
joro's user avatar
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19 votes
1 answer
1k views

Estimate on radical of $2^n \pm 1$

Not sure if this belongs to MO or not. Are there any lower bound on radical of $2^n \pm 1$? We recall that radical of an integer $rad(k)$ is a product of primes which divide $k$. As an example, if ...
Ostap Chervak's user avatar
15 votes
1 answer
2k views

Forbidden polynomial identities by the abc conjecture

The Mason–Stothers theorem states Let $a(t), b(t)$, and $c(t)$ be relatively prime polynomials such that $a + b = c$, with coefficients that are either real numbers or complex numbers. Then $\...
joro's user avatar
  • 24.2k
37 votes
2 answers
3k views

A weaker version of the ABC conjecture

I posted this question over at Stackexchange, where a user informed me that it was probably more appropriate for Mathoverflow. Here's to hoping that the answer is out there: The ABC conjecture states ...
Ryan's user avatar
  • 335
10 votes
0 answers
2k views

Model-theoretic content of Mochizuki's Teichmüller theory papers

I would like to ask what the specific novel model-theoretic (or set-theoretic) techniques, if any, are that Mochizuki uses in his recent series of four papers. Section 3 of Inter-universal Teichmüller ...
TLss's user avatar
  • 151
9 votes
1 answer
2k views

A question related to the abc conjecture

The abc conjecture asserts that whenever $a,b,c$ are pairwise coprime positive integers such that $a + b = c$ and $\epsilon > 0$, there exists a constant $C_\epsilon > 0$ (which depends on $\...
Stanley Yao Xiao's user avatar
5 votes
0 answers
493 views

What are the consequences of allowing the ABC-conjecture $\kappa_{\epsilon}$ to also vary with $\omega(abc)$?

A commonly encountered form of the ABC-conjecture is the following: For all $\epsilon > 0$, there is a constant $\kappa_{\epsilon} > 0$ (depending only on $\epsilon$) such that for all coprime ...
user02138's user avatar
  • 575
10 votes
1 answer
3k views

Implications of the abc conjecture in Arakelov theory

It is apparent that the abc conjecture is deeply related to Arakelov theory. In one direction, it is shown in S. Lang, "Introduction to Arakelov Theory", that a certain height inequality in Arakelov ...
Anweshi's user avatar
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