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Results tagged with nt.number-theory
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user 7092
Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions
42
votes
Collecting alternative proofs for the oddity of Catalan
Taking the fact that Catalan numbers $C_n$ measure the number of binary trees on $n$ nodes, we can find an involution on the set of these trees: choose the lexicographically first node in the tree tha …
- 1,951
15
votes
4
answers
1k
views
Can we get good rational approximations in all residue classes?
The classic Hurwitz theorem for rational approximations (in simplest form; the constant can of course be improved) gives infinitely many approximations $\frac mn$ to an irrational $\alpha$ with $|\fra …
- 1,951
12
votes
Can we get good rational approximations in all residue classes?
Just to make sure that it's not hidden in the comments for future readers, I want to point out that the paper "On the Approximation of Irrational Numbers With Rationals Restricted By Congruence Relati …
8
votes
Accepted
Is there some numerical evidence that $ \pi(x+x^{1/e})-\pi(x)\geq 1 $ for any large enough $...
Turning my comment into an answer: your conjecture is supported by numerical computation, but much stronger ones are also supported: for instance, Cramér's conjecture, based on models of the primes as …
- 1,951
5
votes
Accepted
Error term for prime harmonic
"Mertens' Proof of Mertens' Theorem" suggests that Mertens had an error term of $O\left(\frac1{\ln x}\right)$, though that's not tight; theorem 14 there offers an $O\left(\frac1{\ln^2x}\right)$ uncond …
- 1,951
4
votes
Counting $\mathrm{mod}\:p$ solutions of Diophantine equation in two variables taking $O(p^2)...
At least for fixed degree, the answer appears to be 'no' — it seems to be well-established that the number of roots of a univariate polynomial $g(x)$ of degree $d$ modulo a prime $p$ can be determined …
- 1,951
3
votes
Mathematics of the 24 game
To expand on Kevin's comment (and using an answer since a comment doesn't have enough characters!) : one other obvious-but-relevant constraint that's going to be an issue for large values of $N$ is th …
- 1,951
2
votes
0
answers
53
views
What lower bounds are known on growth of the distribution of the abundancy index?
Let $a(n)=\sigma(n)/n$ be the abundancy index of $n$ and let $F(x)$ be the distribution function of this index: i.e., the proportion of integers $n$ with $a(n)\leq x$. (This function is well-defined a …
- 1,951