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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
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
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
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
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
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 …
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
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