Questions tagged [abc-conjecture]

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Are there at least one of $a$, $b$, $c$, $a+b$, $b+c$, $c+a$ have h(P) $\le 3$? (was checked up to $10^{18}$)

Given a positive integer $P>1$, let its prime factorization be written $$P=p_1^{a_1}p_2^{a_2}p_3^{a_3}\cdots p_k^{a_k}$$ Define the functions $h(P)$ by $h(1)=1$ and $h(P)=\min(a_1, a_2,\ldots,a_k)$...
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A reinterpretation of the $abc$ - conjecture in terms of metric spaces?

I hope it is appropriate to ask this question here: One formulation of the abc-conjecture is $$c < \text{rad}(abc)^2$$ where $\gcd(a,b)=1$ and $c=a+b$. This is equivalent to ($a,b$ being ...
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How small can $u$ be in the Pell equation $u^2-k^3 v^2=\pm 1$?

Let $k$ be positive integer, not a square and let $u_k,v_k$ be non-trivial solutions to the Pell equation $u_k^2-k^3 v_k^2=\pm 1$. Q1 How small $u_k$ can be infinitely often as function $k$? This ...
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Is this counter example of the ABC conjecture? [closed]

My question is realy question, please don't vote down and vote to close! By the Euler's theorem we have: if $n$ and $a$ are coprime positive integers, then $$a^{\varphi (n)} \equiv 1 \pmod{n}$$ ...
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Are there infinitely many primes $p$, positive integers $k, n$ such that $1 \le n < p$ and $p^k > n.rad(p^{k+1}−n)$?

Among $168$ prime numbers in range $1$ to $10^3$, there are $84$ prime numbers $n$ such that: $p^k> n.rad(p^{k+1}−n)$ where $1 \le n<p$ and $k=2,3,4$. There are also $84$ prime numbers $n$ such ...
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Large radical of an integer and three AB conjectures

In this Note, We propose a new definition called "large radical of an integer". Using this definition, three very useful $AB$ conjecture are given. 1. Large counter examples of the ABC conjecture ...
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Is the conjecture $min(A,B) \le rad(ABC)$ new and correct? [closed]

$\DeclareMathOperator\rad{rad}$Conjecture: If $A, B, C$ are positive integers with $\gcd(A, B)=1$, $\gcd(B, C)=1$, and $\gcd(C, A)=1$, and if $A+B=C$, then $\min(A,B) \le \rad(ABC)$. If the ...
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Inspire from the Fermat last theorem to another ABC conjecture

I. INTRODUCTION We know that the Fermat last theorem states that no three positive integers $a$, $b$, and $c$ satisfy the equation $a^n + b^n = c^n$ for any integer value of $n$ greater than $2$. The ...
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A generalization of Lander, Parkin, and Selfridge conjecture

My question: Are the conjectures as follows correct? Given a positive integer $P>1$, let its prime factorization be written $$P=p_1^{a_1}p_2^{a_2}p_3^{a_3}...p_k^{a_k}$$. Define the ...
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abc conjecture and surjective polynomials

Let $a,b$ be coprime multivariate polynomials with integer coefficients and $\deg(a) > \deg(\rm{rad}(a b))$. Let $c=a+b$ and assume $c$ is either surjective or $c$ represents infinitely many ...
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The abc conjecture modulo variety

It is known that the abc conjecture can't fail with polynomial identities. Is the following special case of abc known? Let $a,b,c,f$ be polynomials with integer coefficients satisfying $a+b=c+f$. ...
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Anabelian geometry ~ higher category theory

Note: I'm worried this question might be taken as controversial, because it relates to Shinichi Mochizuki’s work on the abc conjecture. However, my question has nothing to do with the correctness of ...
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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,\$ ...
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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 ...
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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'...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $$a+b = c = \mathrm{rad}(abc)^{q(a,b,c)}.$$...
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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-...
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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 ...
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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 $\varepsilon >0.$ ...
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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 ...
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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 ...
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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 ...
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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: ...
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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 ...
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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 \$\...