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Results tagged with nt.number-theory
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user 48142
Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions
7
votes
Divisibility condition implies $a_1=\dotsb=a_k$?
Here's a a tweak of Seva's idea that gives a counterexample. Note that if $r$ is odd, then $2^{n}+1$ divides $2^{rn} + 1$.
Let $k = 6$, $a_{1} = 1$, $a_{2} = a_{3} = a_{4} = 2$, $a_{5} = a_{6} = 4$. T …
- 19.9k
7
votes
Accepted
Isomorphism between genus 1 modular curves and elliptic curves
The maps $j : X_{0}(N) \to \mathbb{P}^{1}$ are given in Magma's ''Small modular curves'' database. In each case, they construct functions on the modular curves $X_{0}(N)$ out of eta products, modular …
- 19.9k
4
votes
Accepted
Generating DataSet of Strong PseudoPrimes?
I'm guessing you will want to be working with numbers larger than 64-bits, and so you probably want GMP (see this page). This library is used by much of the software that number theorists use. (Magma …
- 19.9k
12
votes
Accepted
Etale cohomology approach on $\tau(n)$
One of the goals of the development etale cohomology was to generate a cohomology theory that could successfully count points on varieties over finite fields, with one of the main goals of proving the …
- 19.9k
6
votes
Approximations to $\pi$
This is not exactly an answer to the stated question, but it's too long for a comment. Rather than the form given in the question, one could represent a number in the form $\frac{a + b \sqrt{d}}{c}$, …
- 19.9k
4
votes
Accepted
Best error terms for functions related to square free numbers
As I say in the comment, the asymptotics for $M_{+}$ and $M_{-}$ follow directly from those for $M$ and $\hat{M}$. Therefore $M_{+}(x) = \frac{1}{2 \zeta(2)} x + \frac{1}{2} M(x) + O(x^{1/2})$ and $M_ …
- 19.9k
7
votes
Accepted
Question about iterations not divisible by infinitely many prime numbers
Yes. This follows from a result of Corrales-Rodrigáñez and Schoof (see the paper here) solving the support problem of Erdős.
In particular, suppose that there are only finitely many primes $p$ that do …
- 19.9k
22
votes
Accepted
On Fibonacci numbers that are also highly composite
The largest highly composite Fibonacci number is $F_{3} = 2$.
If $p$ is a prime number, then either $p \mid F_{p-1}$ (if $p \equiv \pm 1 \pmod{5}$), $p \mid F_{p}$ (if $p = 5$), or $p \mid F_{p+1}$ (i …
- 19.9k
34
votes
Accepted
Question on a generalisation of a theorem by Euler
I suspect that $k = 4$ is good, but am not sure how to prove it. However, every positive integer $k \geq 5$ is good. This follows from the fact (see the proof of Theorem 1 from this preprint) that for …
- 19.9k
10
votes
Accepted
Prime numbers in a sparse set
Yes, there is a $c > 1$ for which infinitely many numbers of the form $\lfloor k^{c} \rfloor$ are prime. The first result of this type was proven in Ilya Piatetski-Shapiro's Ph.D. thesis (written in 1 …
- 19.9k
17
votes
Accepted
A divisor sum congruence for 8n+6
The congruence you state is true for all $m \equiv 6 \pmod{8}$. The proof I give below relies on the theory of modular forms. First, observe that
$$
\sum_{k=1}^{m-1} d(k) d(m-k) = 2 \sum_{k=1}^{\frac{ …
- 19.9k
9
votes
Accepted
Complexity of computing the number of visible points
There is an algorithm for computing $F(N) = \# \{ (a,b) : 1 \leq a, b \leq N, \gcd(a,b) = 1 \}$ in time $O(N^{5/6 + \epsilon})$. This relies on the algorithm of Deleglise and Rivat (see their paper he …
- 19.9k
7
votes
The Chebotarev Density Theorem and the representation of infinitely many numbers by forms
The example of Heath-Brown's article is a good one. For a bit more elementary examples, you can fix a number field $K$ with $[K : \mathbb{Q}] = n$ and ring of integers $\mathcal{O}_{K}$ and pick a bas …
- 19.9k
3
votes
Minimum number of unit fractions to sum up a given positive rational
No such polynomial-time algorithm exists because in some instances it would take too long to write down (or store in memory) the answer. In particular, if $n$ is a positive integer, then we have $\sum …
- 19.9k
10
votes
Accepted
Extension of a formula for the quadratic Gauss sums
No, the relation is not as simple in this case. For example, if $k = 3$ and $p = 7$, the three different cubic Gauss sums are roots of $y^{3} - 21y - 7$, and the three roots of this polynomial do not …
- 19.9k