Found problems: 721
Find all triplets of prime numbers $(m, n, p)$, that satisfy the system of equations: $$\left\{\begin{matrix} 2m-n+13p=2072,\\3m+11n+13p=2961.\end{matrix}\right.$$
For a nonnegative integer $n$ define $\operatorname{rad}(n)=1$ if $n=0$ or $n=1$, and $\operatorname{rad}(n)=p_1p_2\cdots p_k$ where $p_1<p_2<\cdots <p_k$ are all prime factors of $n$. Find all polynomials $f(x)$ with nonnegative integer coefficients such that $\operatorname{rad}(f(n))$ divides $\operatorname{rad}(f(n^{\operatorname{rad}(n)}))$ for every nonnegative integer $n$.
Determine all prime numbers $p$ for which $8p^4-3003$ is a positive prime number.
Let $p \ge 2$ be a prime number and $a \ge 1$ a positive integer with $p \neq a$. Find all pairs $(a,p)$ such that: $a+p \mid a^2+p^2$
Let $n$ and $k$ be two positive integers such that $1\leq n \leq k$. Prove that, if $d^k+k$ is a prime number for each positive divisor $d$ of $n$, then $n+k$ is a prime number.
Is it possible to place 100 consecutive numbers around a circle in some order such that the product of each pair of adjacent numbers is always a perfect square? (Recall that a number is a perfect square if it is the square of an integer.)
Prove that if a prime is the sum of four perfect squares then the product of two of these is equal to the product of the other two.
[i]Gherghe Stoica[/i]
Show that there exists a set $ A$ of positive integers with the following property: for any infinite set $ S$ of primes, there exist [i]two[/i] positive integers $ m$ in $ A$ and $ n$ not in $ A$, each of which is a product of $ k$ distinct elements of $ S$ for some $ k \geq 2$.
Let $p$ be a prime number for which $\frac{p-1}{2}$ is also prime, and let $a,b,c$ be integers not divisible by $p$. Prove that there are at most $1+\sqrt {2p}$ positive integers $n$ such that $n<p$ and $p$ divides $a^n+b^n+c^n$.
Determine all prime numbers $p$ such that $p^2 - 6$ and $p^2 + 6$ are both prime numbers.
Find all distinct prime numbers $p,q,r,s$ such that $1-\frac{1}{p} - \frac{1}{q} -\frac{1}{r} - \frac{1}{s} =\frac{1}{pqrs}$
Let $n\ge 2$ be a positive integer. For any integer $a$, let $P_a(x)$ denote the polynomial $x^n+ax$. Let $p$ be a prime number and define the set $S_a$ as the set of residues mod $p$ that $P_a(x)$ attains. That is, $$S_a=\{b\mid 0\le b\le p-1,\text{ and there is }c\text{ such that }P_a(c)\equiv b \pmod{p}\}.$$Show that the expression $\frac{1}{p-1}\sum\limits_{a=1}^{p-1}|S_a|$ is an integer.
[i]Proposed by fattypiggy123[/i]
Find all prime numbers $p$ such that there exists a unique $a \in \mathbb{Z}_p$ for which $a^3 - 3a + 1 = 0.$
A twin prime pair is a pair of primes $(p,q)$ such that $q = p + 2$. The Twin Prime Conjecture states that there are infinitely many twin prime pairs. What is the arithmetic mean of the two primes in the smallest twin prime pair? (1 is not a prime.)
$\textbf{(A) }4$
Let $p$ be a prime number and let $A$ be a set of positive integers that satisfies the following conditions:
(i) the set of prime divisors of the elements in $A$ consists of $p-1$ elements;
(ii) for any nonempty subset of $A$, the product of its elements is not a perfect $p$-th power.
What is the largest possible number of elements in $A$ ?
Prove that for all $a, b$ and $x_0$ positive integers, in the sequence $x_1, x_2, x_3, \cdots$ defined by
$$x_{n+1}=ax_n+b, n\geq 0$$
Exist an $x_i$ that is not prime for some $i\geq 1$
Check if there exists positive integers $ a, b$ and prime number $p$ such that $a^3-b^3=4p^2$
Let define a function $f: \mathbb{N} \rightarrow \mathbb{Z}$ such that :
$i)$$f(p)=1$ for all prime numbers $p$.
$ii)$$f(xy)=xf(y)+yf(x)$ for all positive integers $x,y$
find the smallest $n \geq 2016$ such that $f(n)=n$
Two different prime numbers between $ 4$ and $ 18$ are chosen. When their sum is subtracted from their product, which of the following numbers could be obtained?
$ \textbf{(A)}\ 21 \qquad \textbf{(B)}\ 60\qquad \textbf{(C)}\ 119 \qquad \textbf{(D)}\ 180\qquad \textbf{(E)}\ 231$
An ordered triple $(p, q, r)$ of prime numbers is called [i]parcera[/i] if $p$ divides $q^2-4$, $q$ divides $r^2-4$ and $r$ divides $p^2-4$. Find all parcera triples.
Prove that there is an integer $n>10^{2018}$ such that the sum of all primes less than $n$ is relatively prime to $n$.
[i](R. Salimov)[/i]
$\mathbb{N}$ is the set of positive integers. Determine all functions $f:\mathbb{N}\to\mathbb{N}$ such that for every pair $(m,n)\in\mathbb{N}^2$ we have that:
\[f(m)+f(n) \ | \ m+n .\]
Let $m, n$ be odd prime numbers.
Find all pairs of integers numbers $a, b$ for which the system of equations:
$x^m+y^m+z^m=a$,
$x^n+y^n+z^n=b$
has many solutions in integers $x, y, z$.
A positive integer $ n$ is called $ good$ if it can be uniquely written simultaneously as $ a_1\plus{}a_2\plus{}...\plus{}a_k$ and as $ a_1 a_2...a_k$, where $ a_i$ are positive integers and $ k \ge 2$. (For example, $ 10$ is good because $ 10\equal{}5\plus{}2\plus{}1\plus{}1\plus{}1\equal{}5 \cdot 2 \cdot 1 \cdot 1 \cdot 1$ is a unique expression of this form). Find, in terms of prime numbers, all good natural numbers.
Find all prime numbers $p$ and all positive integers $n$, such that
$n^8 - n^2 = p^5 + p^2$