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Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions
7
votes
Primes dividing x^4 -2
You want an elementary argument for the existence of an infinite set of primes $p$ which at once divide a number of the form $x^4-2$ and a number of the form $y^4+1$, where $x,y \in \mathbb{Z}$. Here …
1
vote
Generalization of "Hadamard quotient theorem" to higher genus and positive equicharacteristic?
First, I am not sure how appropriate it is to answer my own question. If this is against the protocol here, let me know, and I'll promptly delete this answer and copy it as an addendum instead.
Here …
3
votes
The distribution of fractional parts $\Big\{ \frac{N}{n} \Big\}$
As $N \to \infty$, the set of fractional parts
$$
\Big\{ \frac{N}{n}\Big\}, \quad 1 \leq n < \sqrt{N}
$$
becomes asymptotically equidistributed in the Lebesgue measure of $[0,1]$. The same persists w …
9
votes
Accepted
An effective version of Kronecker's approximation theorem and its variations
The numbers $\theta_i$ here are real of course. Also there is only one question really; upon replacing $\boldsymbol{\alpha}$ with $(\boldsymbol{\alpha} - t \boldsymbol{\theta})/s$ the second problem r …
8
votes
0
answers
512
views
How many curves in a family possess a rational point?
Consider $B$ a finite-type integral quasi-projective scheme over $\mathbb{Z}$ such that $B(\mathbb{Z})$ infinite. (If you like, take $B$ to be the affine line). Let $X \to B$ be a generically smooth p …
8
votes
1
answer
461
views
The critical exponent in the multiplicative order of 2 modulo primes
This is a sequel to this MO question:
The multiplicative order of 2 modulo primes
As shown in Charles Matthews' paper linked to there, it is not hard to show that for each $\delta > 0$ there is a $ …
8
votes
0
answers
503
views
A refinement of Lehmer's conjecture?
Added. (24/8) Let me add another motivation by phrasing the question in a much bolder (perhaps too optimistic?) form, in which a weaker estimate may actually be proved. Compare this to the following s …
5
votes
1
answer
734
views
Generalization of "Hadamard quotient theorem" to higher genus and positive equicharacteristic?
Added. (9/8/2013). Here is an explicit case. Recall that the series $\sum_{n \geq 0} \binom{2n}{n}t^n = (1-4t)^{-1/2}$ is algebraic. Certainly there are polynomials $f(n)$, all of which split into lin …
9
votes
1
answer
330
views
What is the set of possible densities of pointless members in a family of rational curves ov...
This is a sequel to the following previous MathOverflow posts, but the question itself appears to be of a different flavor, being limited to rational curves:
Are most curves over Q pointless?
How m …
63
votes
3
answers
5k
views
Are there infinitely many integer-valued polynomials dominated by $1.9^n$ on all of $\mathbb...
The original post is below. Question 1 was solved in the negative by David Speyer, and the title has now been changed to reflect Question 2, which turned out to be the more difficult one. A bounty of …
8
votes
0
answers
217
views
Attractors of arithmetically small points
Consider the "points" to be in $\mathbb{G}_m(\mathbb{C}) = \mathbb{C}^{\times}$. By a sequence of "arithmetically small points" is meant a sequence of pairwise different algebraic points $\beta$ that, …
26
votes
irreducibility of discriminant
The discriminant locus has the following geometric interpretation, given in the introductory chapter of [Gelfand, Kapranov, Zelevinsky: Discriminants, Resultants and Multidimensional Determinants].
L …
5
votes
Accepted
Reducibility of resultants
All such resultants and/or discriminants are geometrically irreducible in characteristic zero, and a power of an irreducible in general. This is actually covered by the geometric argument I quoted in …
2
votes
Small values of a polynomial evaluated at roots of unity
As in Baker's theorem $|\prod_i \alpha_i^{n_i}-1| \gg n^{-C}$, $n := \max_i{|n_i|}$, this would be another instance where the trivial Liouville lower bound (exponential in $-n$) should actually be rep …
1
vote
Small values of a polynomial evaluated at roots of unity
After a couple of failed attempts at a proof, I have come to appreciate the difficulty of even the subexponential bound $e^{-o(n)}$.
The lemma I had asserted does not follow from the Theorem below if …