0
votes
0answers
30 views

Differential Diophantine Equations?

So this is both a question on its own as well as a request for where I can find information on a given topic. Consider Differential Equations in two variables of the form: $$P(Z,Z', Z'' ... Z^{[n]}, ...
1
vote
0answers
61 views

Are even Dirichlet series constants?

I consider a Dirichlet series with an absolute abscissa of convergence $\sigma$ that can be meromorphically extend to $\mathbb{C}$: $$\phi(s)=\sum_{n=1}^{+\infty}{\frac{a_n}{n^s}}$$ and the analytic ...
0
votes
1answer
47 views

Functional equations and cubes

Problem $10728$ from Amer. Math. Monthly "Preserving the sum of three cubes" says: Determine all functions $f:\mathbb{Z}\rightarrow \mathbb{Z}$ satisfying $f(x^3+y^3+z^3)=(f(x))^3+(f(y))^3+(f(z))^3$ ...
10
votes
1answer
329 views

A unsolved puzzle from Number Theory/ Functional inequalities

The function $g:[0,1]\to[0,1]$ is continuously differentiable and increasing. Also, $g(0)=0$ and $g(1)=1$. Continuity and differentiability of higher orders can be assumed if necessary. The ...
3
votes
2answers
95 views

Finding every possible $f(x), a\in\mathbb R$ such that ${\{f(x)\}}^2=a+f(x^2)$

Let $f(x)$ be a rational expression of $x$, and let $a$ be a real number. Then, I'm facing difficulty for finding every possible $a, f(x)$ such that $${\{f(x)\}}^2=a+f(x^2).$$ Here, suppose that ...
5
votes
3answers
108 views

If x and y are different integers , and if $2005 +x =y^2 ; 2005+y =x^2 $ then find xy…

Problem : If $2005 +x =y^2 ; 2005+y =x^2$ then find xy... My approach : Let $2005 +x =y^2 .....(i) ; 2005+y =x^2 ......(ii) $ Now from (i) we get : $ y = \sqrt{x + 2005}$ Now putting this ...
2
votes
2answers
47 views

why is it constant?

$f$ is a function from $\Bbb R$ to $\Bbb R$: $\frac{f(x+y)}{(x+y)} - (x+y)^2 = \frac{f(x-y)}{(x-y)} - (x-y)^2$ for all $x$ and $y$ the solution book just says "Thus $\frac{f(x)}{x} - x^2$ is a ...
1
vote
1answer
81 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* ...
7
votes
0answers
56 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
496 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
112 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
509 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
95 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
296 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 $P(x)=x+1$.