The abc-conjecture tag has no usage guidance.

**28**

votes

**2**answers

833 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 \\
...

**15**

votes

**3**answers

657 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:
...

**10**

votes

**0**answers

448 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 = ...

**10**

votes

**1**answer

661 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 $2^n \pm 1$ (Recall that radical of integer $rad(k)$ is a product of primes which divide integer $k$)?
As an example If ...

**9**

votes

**3**answers

789 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 + ...

**9**

votes

**1**answer

2k 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 ...

**9**

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 ...

**8**

votes

**1**answer

638 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 ...

**8**

votes

**3**answers

810 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 ...

**6**

votes

**1**answer

1k 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 ...

**5**

votes

**0**answers

133 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

**2**answers

391 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 ...

**4**

votes

**0**answers

216 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

240 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

133 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

173 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 ...

**0**

votes

**1**answer

804 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: ...