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Results tagged with nt.number-theory
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user 11926
Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions
7
votes
Accepted
Generalizing Ramanujan's "1729 story"
There are many articles that study the quantity you call $r_n(k)$ using sieve methods. Among them I mention the following, which give highly non-trivial bounds for the number of $k<X$ such that $r_n(k …
- 45.3k
4
votes
sum of four squares with some coefficients
First, one doesn't normally call $(a,b,c,d)$ a "pair" of rational numbers. In any case, first check that your equation has a solution with $(b,d)\ne0$ in $\mathbb R$ (which it clearly does) and in $\m …
- 45.3k
5
votes
Have you ever seen this product?
Elementary comment. For a fixed real number $x$, define a multiplicative function $f(x;n)$ as follows. Factor $n=p_1^{e_1}\cdots p_r^{e_r}$ and set $f(x;n)=x^{e_1+\cdots+e_r}$. Then
$$
\prod \left(1 …
- 45.3k
3
votes
Divisibility among discriminants
There are a number of papers that deal with fields generated by points in the inverse image of iterates. So in your setting, let $c=f(f(r))=f^2(r)$, then $f(r)$ is in $f^{-1}(c)$, the first inverse im …
- 45.3k
10
votes
Quadratic squares
A non-trivial answer for your comment/question for an example giving $\Omega(\log N)$, take $p(x)=2x^2+1$. This gives a Pell equation $2x^2+1=y^2$, and taking powers of the fundamental unit will, I be …
- 45.3k
10
votes
Effective Mordell
If one had an effective version of the abc conjecture, then Elkies showed how to use it to obtain an effective version of the Mordell conjecture (using Belyi maps).
In your formulation, the theorem w …
- 45.3k
7
votes
Accepted
Given any $(a,m,n)\in \mathbb{Z}\times\mathbb{N}^2$ with $\gcd(a,m)=1$ is there a quick way ...
The Goldwasser-Micali probabilistic cryptosystem is based on exactly this principle. Let $N=pq$ and let $a$ be an integer with $\left(\frac{a}{p}\right)=\left(\frac{a}{q}\right)=-1$, i.e., $a$ is a no …
- 45.3k
3
votes
Accepted
Bounded differences in exponential sequences
For any fixed integers $a$ and $b$ that are multiplicatively independent and for any integer $k$, there are only finitely many pairs of positive integers $m$ and $n$ such that
$$ a^n - b^m = k. $$
To …
- 45.3k
4
votes
Computing a polynomial product over roots of unity
The transformation
$$
\mathcal{L}_n : \mathbb{C}(x) \longrightarrow \mathbb{C}(x)
$$
defined by
$$
\mathcal{L}_n(F(x)) = \sum_{\zeta\in\boldsymbol\mu_n} F(\zeta y)\Big|_{y^n\to x}
$$
is called a …
- 45.3k
5
votes
Conjecture on Markov-Hurwitz Diophantine equation
Not an answer, but a bit long for a comment. If you don't already know them, you might find the papers of Baragar to be of interest:
[1] The exponent for the Markoff-Hurwitz equations. Pacific J. Mat …
- 45.3k
9
votes
Accepted
Local-to-global principle for certain genus 0 curves
The former always has rational solutions. Let $x=tu$, $y=tv$, $z=tw$, then
$$ au^2 + bv^2 = c t w^3. $$
So simply choose any $u,v,w\in\mathbb Q^*$ that you want, set
$$ t = \frac{au^2+bv^2}{cw^3}, $$
…
- 45.3k
3
votes
A question on degree 4 binary forms
Have you left some condition off of Stewart's conjecture? It's not right as you've stated it. Indeed:
Theorem Let $f(x,y)\in\mathbb{Z}[x,y]$ be a polynomial of degree $3$ with $\operatorname{Disc}(f …
- 45.3k
8
votes
Accepted
Sequences satisfying gcd(S(x), S(y)) = S(gcd(x,y))
A strong divisibility sequence is a sequence of positive integers $(a_n)_{n\ge1}$ with the property that $\gcd(a_n,a_m)=a_{\gcd(n,m)}$. See http://en.wikipedia.org/wiki/Divisibility_sequence.
(Strong) …
- 45.3k
5
votes
When are infinitely many points in the orbit of a polynomial integers?
A slight variant that turns out to be non-elemenatry is to replace the polynomial with a rational function. This leads to:
Theorem: Let $R(x)\in\mathbf{Q}(x)$ be a rational function of degree at leas …
- 45.3k
5
votes
Intersection of $\{2^a 3^b 5^c 7^d\}$ and its translates
As Mike Bennett said, this is an example of an $S$-unit equation, although you are asking for solution to $u-v=1$ in $\mathbb Z_S^*\cap \mathbb Z$. More generally, one drops the requirement that $u$ a …
- 45.3k