The abc-conjecture tag has no wiki summary.

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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 ...

**1**

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155 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 ...

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63 views

### How small the radical of ternary form of degree > 3 can be infinitely often?

This is related to identities for the abc conjecture and
to Granville-Langevin conjecture. Generalization of GL fails for
3 or more variables.
Let $f \in \mathbb{Z}[x,y,z],\deg(f) > 3$ be ...

**14**

votes

**3**answers

569 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:
...

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178 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 ...

**28**

votes

**2**answers

731 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 \\
...

**3**

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120 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

763 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 ...

**9**

votes

**3**answers

727 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 + ...

**4**

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315 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 ...

**5**

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 ...

**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 ...

**9**

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**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 ...