Tagged Questions
1
vote
1answer
64 views
Solution to a functional equation
Let $n,i$ be positive integers and $C$ a strictly positive real value.
Consider the equation for $f$ :
$$1*\ln(f(n)) = C * \sum_{3<i* \ln(i) < \sqrt{n}} \left(\ln[f( i* \ln(i) )-1)] - \ln[(f( i* ...
4
votes
0answers
28 views
What are the densities of branches of the euclidean tree?
The Euclidean algorithm shows how all coprime pairs of positive integers can be uniquely obtained from the pair $(1,1)$ by applying the two operations $(a,b) \to (a+b,b)$ and $(a,b) \to (a,a+b)$.
(or ...
2
votes
2answers
258 views
Functional Equation. $f(mn)=f(m)f(n)$ and …
I want to prove the following.
We have a function $f: \mathbb{Z} \to\mathbb{R}$ s.t.
(1) $f(mn) = f(m)f(n)$
(2) $f(m+n) \leq f(m) + f(n)$
(3) $0 \leq f(x) \leq 1$
then $f(m+n) \leq \max\big(f(m), ...
4
votes
2answers
104 views
How many ways to reach $1$ from $n$ by doing $/13$ or $-7$?
How many ways to reach $1$ from $n$ by doing $/13$ or $-7$ ?
(i.e., where $n$ is the starting value (positive integer) and $/13$ means division by $13$ and $-7$ means subtracting 7)?
Let the number ...
7
votes
2answers
383 views
Problem on Euler's Phi function
Let $S(n)$ be $S(n)=\left\{k\;\left|\;\left\{\frac{n}{k}\right\}\right.\geq \frac{1}{2}\right\}$,where $\{x\}$ is the fractional part of $x$
Prove that :
\begin{align}
\sum_{k\in S(n)} ...
2
votes
1answer
86 views
All functions with the property $ k \mid f(m+n) \iff k \mid f(m)+f(n)$
Let $\mathbb N$ be the set of all positive integers. How can one find all functions $f: \mathbb N \to \mathbb N$ such that
$$ k \mid f(m+n) \iff k \mid f(m)+f(n)$$
For all positive integers $k$.
6
votes
1answer
221 views
Iterated polynomial problem
Polynomial $P$ satisfies $P(n)>n$ for all positive integers $n$. Every positive integer $m$ is a factor of some number of the form $P(1),P(P(1)),P(P(P(1))),\ldots $. Prove that $P(x)=x+1$.
