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**16**

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**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
$$a+b = c = \mathrm{rad}(abc)^{q(a,b,c)}.$$...

**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 ...

**5**

votes

**0**answers

137 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 $\varepsilon >0.$ ...

**4**

votes

**0**answers

259 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 ...

**2**

votes

**0**answers

257 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 $\...

**2**

votes

**0**answers

142 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 $...

**1**

vote

**0**answers

187 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 ...

**-1**

votes

**0**answers

70 views

### Sum-free sets of powerful numbers

For $n=p_1^{\alpha_1}\cdots p_r^{\alpha_r}$ with distinct primes $p_i$, call $\alpha= (\alpha_1,\dots,\alpha_r)$ the type of $n$ and denote by $N_\alpha$ the set of all naturals of this type.
We ...