Questions tagged [abc-conjecture]
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84
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On two "versions" of abc conjecture
Let $a,b,c$ be coprime nonzero positive integers such that $a+b=c$. The ABC conjecture states that for any $\varepsilon>0$, we have $$c < C_{\varepsilon}\operatorname{rad}(abc)^{1+\varepsilon}.$$...
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2
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1
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241
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Small solutions of $x^2-a^3 y^2=\pm 1$
We are interested in small integer solutions to the Pell equation:
$$x^2-a^3 y^2=\pm 1 \qquad (1)$$
Where in $\pm 1$ you can chose either sign.
$(x^2,a^3 y^2)$ are consecutive powerful numbers.
$abc$ ...
- 24.7k
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Is there a mistake in Mochizuki's proof of Theorem 1.10 in IUTT IV? [closed]
In Global character of ABC/Szpiro inequalities, Peter Scholze says that he thinks Joshi's proof of the abc conjecture in his paper has a mistake in Proposition 6.10.7. However, for the proof of ...
- 2,574
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Global character of ABC/Szpiro inequalities
In [M24] it is asserted that "considering $abc$ triples of the form $(1,p^n,1+p^n)$ for a prime number $p$ and an arbitrarily large positive integer $n$, one can verify that ABC/Szpiro ...
- 869
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Some specific case of the abc-conjecture related to the Mason-Stothers proof of Serge Lang for polynomials?
I stumbled upon a version of the Mason-Stothers theorem, which can be modified to make a similar statement for natural numbers, which I think I can prove here. It is not the abc-conjecture, but it has ...
- 2,608
3
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1
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518
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abc-conjecture and positive definite kernels, again?
One formulation of the abc-conjecture is:
$$\forall a,b \in \mathbb{N}: \frac{a+b}{\gcd(a,b)}< \operatorname{rad}\left ( \frac{ab(a+b)}{\gcd(a,b)^3}\right )^2 $$
Let us define:
$$K(a,b) := \frac{2(...
- 2,608
2
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110
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On a subset of the $abc$ triples
The $abc$ conjecture states that, for every positive real $\varepsilon$, there exist only finitely many triples $(a, b, c)$ of coprime positive integers such that $a + b = c$ and
$$c > \...
62
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6
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Consequences of Kirti Joshi's new preprint about p-adic Teichmüller theory on the validity of IUT and on the ABC conjecture
Today, somebody posted on the nLab a link to Kirti Joshi's preprint on the arXiv from last month: https://arxiv.org/abs/2210.11635
In that preprint, Kirti Joshi claims that
he agrees with Scholze and ...
- 2,574
3
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390
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Simple Diophantine equation
Are there any solutions in positive integers of
$x^3 + 1 = (x - k) y^3$?
The closest I can get is
$19^3 + 1 = 20 \times 7^3$,
but $20\gt 19$ so it just misses!
For the related
$x^3 - 1 = (x - k) y^3$,...
- 827
2
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264
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Baby $abc$ conjecture for $n$-th roots
Is there any progress on a “baby $abc$ conjecture” where you restrict attention to rational approximations of $n$-th roots?
Let $r/s$ be a very close approximation to $(t/u)^{1/n}$, so that
$$
|u\cdot ...
- 827
7
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0
answers
447
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Is it unconditionally known that abc conjecture can't fail on a variety?
Background: this question gives the identity:
$$(x+z)^5+(y-z)^5 = (-3 x + 4 y)^2 (x + y)^3 + (x+y) f(x,y,z)$$
The curve $C : f(x,y,z)=0$ is genus 1, have infinitely many rational and integral points ...
- 24.7k
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273
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Does $abc$ preclude very smooth solutions?
Recall the $abc$-conjecture, which asserts that for any $\epsilon > 0$ there exists a positive number $C(\epsilon)$ such that for any coprime integers $a,b,c$ with $a + b = c$ and $\max\{|a|, |b|, |...
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Ruling out an extremely specific class of Wieferich-like primes
Recall that a prime $p$ is a Wieferich prime if $p^2|2^{p-1}-1$. The only known Wieferich primes are $p=1093$ and $p=3511$. A prime $p$ is a generalized Wieferich prime to base $q$ if $p^2|q^{p-1}-1$.
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The average power of an integer and a strengthened Fermat Last Theorem
It is well known that the ABC conjecture gives an immediate proof of Fermat Last Theorem (FLT). It seems that it proves something stronger involving not necessarily perfect power, which may still be ...
- 1,202
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1
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390
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Is this set dense in [0,+∞)?
We define $A=\{ \frac{c}{rad(abc)}: a, b > 0, c=a+b, gcd(a, b)=1 \}$.
Is the set $A$ dense in $[0, +\infty)$?
Does $\overline{A}$ have interior? Here $\overline{A}$ is the closure of $A$.
A well-...
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