Questions tagged [abc-conjecture]
The abc-conjecture tag has no usage guidance.
80
questions
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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 ...
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 \...
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/...
1
vote
1
answer
282
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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 ...
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 ...
5
votes
1
answer
2k
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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
votes
1
answer
3k
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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
1
answer
4k
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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
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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
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
$$a+b = c = \mathrm{rad}(abc)^{q(a,b,c)}.$$...
0
votes
1
answer
3k
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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-...
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 ...
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 $\varepsilon >0.$ ...
9
votes
1
answer
908
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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 ...
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 ...
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 ...
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:
...
2
votes
0
answers
381
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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 $\...
35
votes
2
answers
2k
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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(\...
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 $...
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-\...
10
votes
3
answers
1k
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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 + ...
19
votes
1
answer
1k
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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 ...
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 $\...
37
votes
2
answers
3k
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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 ...
10
votes
0
answers
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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 ...
9
votes
1
answer
2k
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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 $\...
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 ...
10
votes
1
answer
3k
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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 ...