Comment by Command Master on for which $A\subset \mathbb{N}$ does there exist...
Intuitively it seems to me that almost all sequences should work for all numbers, even if you make them grow exponentially
View ArticleComment by Command Master on Maximizing a sum minus its maximal summand
You might want to fix (an upper bound to) $\max i\pi_i$ and then maximize $\sum i\pi_i$ as a function of that, and then maximize that.
View ArticleComment by Command Master on Is Reciprocal Fibonacci constant irrational?
The reciprocal prime Fibonacci constant being irrational implies there are infinitely many Fibonacci primes, which is an open problem.
View ArticleComment by Command Master on A question about the prime counting function
Is there a reason this question is framed in terms of $n+1$ and not $n$ ($\pi(n^2) < \frac{(n+1)^2}{\ln((n+1)^2)}$)?
View ArticleComment by Command Master on Approximating a fraction with a given denominator
@mtheorylord There are many algorithms for this — just search "two variable jnteger programming"
View ArticleComment by Command Master on Does every real number $r\in [0,1]$ have a...
@JoachimKönig Thanks, I was wondering if that's the case
View ArticleComment by Command Master on How to define a fractal from the lexicographic...
How is the factorization ordered? Are the primes in increasing order? The prime powers? Something else?
View ArticleComment by Command Master on In what sense does the sentence...
The notion of omega-consistency might be relevant here - while there are (consistent) models of PA where $\neg \mathsf{con}(\mathsf{PA})$ holds, they aren't $\omega$-consistent.
View ArticleComment by Command Master on Linear equation among divisors of a positive...
Intuitively this feels too restricted, perhaps the abc or n conjecture can show something
View ArticleComment by Command Master on Why should we expect this odd behavior of...
It might be due to the fact that it only takes integers — can you try to graph the fractional part of $nq/p +2\sqrt{nq}/p$?
View ArticleComment by Command Master on Repeated values of a monomial
Given two solutions $(h_1, m_1), (h_2, m_2)$, you have $\log(\frac{m_1}{m_0}) \log(\frac{h_2}{h_0}) = \log(\frac{m_2}{m_0})\log(\frac{h_1}{h_0})$. There might be something simple this implies which I'm...
View ArticleComment by Command Master on Factorization trees and (continued) fractions?
@mathoverflowUser Each non-leaf node must have a leaf as a child, and that leaf has a value of $1$, so in $p(T_{n,m}, 1)$ the numerator is $1$ and the denominator is at least $1$, so $p(T_{n,m}, 1)...
View ArticleComment by Command Master on Is it possible to write any Dirichlet series as...
Are you talking about Dirichlet series in general, or those of Dirichlet characters?
View ArticleComment by Command Master on Is there a "halting machine" which halts on itself?
@EmilJeřábek thanks, I updated it. A quine is a program which outputs itself, so that's how I remembered it
View ArticleAnswer by Command Master for Expected sorting time of random permutation...
Here's a proof that $\mathbf{E}[X] = O(n^2 \log^2(n))$. Let's assume WLOG $n$ is a power of 2. Let's calculated the expected time until all values bigger than $\frac{n}2$ are in the second half (and...
View ArticleExpected sorting time of random permutation using random comparators
In sorting networks, a comparator of positions $i < j$ is an operator which takes a permutation, checks if $p_i > p_j$, and if it is the case, swaps $p_i$ and $p_j$.Using this, we can define the...
View ArticleAsymptotic number of "modular primes"
We can say that a number $p$ is prime modulo $N$ if for any two numbers $1<a,b<p$, $ab \not\equiv p \pmod N$. We will define $p(n)$ to be the number of primes mod $n$. I'm wondering about the...
View ArticleAnswer by Command Master for Residues distribution modulo an interval
Note that $n \equiv b-1 \pmod b$ implies $b | n+1$, $n \equiv b-2 \pmod b$ implies $b | n+2$, and so on. Therefore you can factor $n+1$, $n+2$, and so on, until one of them has a factor in $I$. You can...
View ArticleEfficiently count the number of primitive roots in all moduli up to $n$
Let's define $f(n)$ as the number of primitive roots modulo $n$. That is, $f(n) = \begin{cases}\varphi(\varphi(n))&n=1,2,4,p^k,2p^k\\0&\text{otherwise}\end{cases}$. We want to efficiently...
View ArticleAnswer by Command Master for Illumination from visible lattice points with...
Let's look at the number of the representations of $n$ as $a^2 + b^2$ with coprime $a, b$, and denote it $v(n)$. Our sum is then $\sum_{n \leq r^2} {\frac{v(n)}{n}}$.We can see that $\sum_{d^2 | n}...
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