Found problems: 1048
Prove that there do not exist distinct prime numbers $p$ and $q$ and a positive integer $n$ satisfying the equation $p^{q-1}- q^{p-1}=4n^3$
Find all natural numbers $a$, $b$, $c$ and prime numbers $p$ and $q$, such that:
$\blacksquare$ $4\nmid c$
$\blacksquare$ $p\not\equiv 11\pmod{16}$
$\blacksquare$ $p^aq^b-1=(p+4)^c$
Determine the greatest common divisor of the numbers $p^6-7p^2+6$ where $p$ runs through the prime numbers $p \ge 11$.
Determine the natural number $a =\frac{p+q}{r}+\frac{q+r}{p}+\frac{r+p}{q}$ where $p, q$ and $r$ are prime positive numbers.
Let $f(x) = x^{1998} - x^{199}+x^{19}+ 1$. Prove that there is an infinite set of prime numbers, each dividing at least one of the integers $f(1), f(2), f(3), f(4), ...$
How many of the first ten numbers of the sequence $121$, $11211$, $1112111$, ... are prime numbers?
$\textbf{(A) } 0 \qquad \textbf{(B) }1 \qquad \textbf{(C) }2 \qquad \textbf{(D) }3 \qquad \textbf{(E) }4$
Let $p{}$ be a prime number and $x_1,x_2,\ldots,x_p$ be integers for which $x_1^n+x_2^n+\cdots+x_p^n$ is divisible by $p{}$ for any positive integer $n{}$. Prove that $x_1-x_2$ is divisible by $p{}.$
Let $a$ be a fixed positive integer and $(e_n)$ the sequence, which is defined by $e_0=1$ and
$$ e_n=a + \prod_{k=0}^{n-1} e_k$$
for $n \geq 1$.
Prove that
(a) There exist infinitely many prime numbers that divide one element of the sequence.
(b) There exists one prime number that does not divide an element of the sequence.
(Theresia Eisenkölbl)
Let $n$ be a given number greater than 2. We consider the set $V_n$ of all the integers of the form $1 + kn$ with $k = 1, 2, \ldots$ A number $m$ from $V_n$ is called indecomposable in $V_n$ if there are not two numbers $p$ and $q$ from $V_n$ so that $m = pq.$ Prove that there exist a number $r \in V_n$ that can be expressed as the product of elements indecomposable in $V_n$ in more than one way. (Expressions which differ only in order of the elements of $V_n$ will be considered the same.)
A prime number$ q $is called[b][i] 'Kowai' [/i][/b]number if $ q = p^2 + 10$ where $q$, $p$, $p^2-2$, $p^2-8$, $p^3+6$ are prime numbers. WE know that, at least one [b][i]'Kowai'[/i][/b] number can be found. Find the summation of all [b][i]'Kowai'[/i][/b] numbers.
Is it possible to choose five different positive integers so that the sum of any three of them is a prime number?
Prove that the number $\lfloor (2+\sqrt5)^{2019} \rfloor$ is not prime.
Are there positive real numbers $a$ and $b$ such that $[an+b]$ is prime for all natural values of $n$ ?
$[x]$ denotes the integer part of the number $x$, the largest integer that does not exceed $x$.
Let $n$ be a positive integer. If $4^n + 2^n + 1$ is a prime, prove that $n$ is a power of three.
Let $Q$ be a set of prime numbers, not necessarily finite. For a positive integer $n$ consider its prime factorization: define $p(n)$ to be the sum of all the exponents and $q(n)$ to be the sum of the exponents corresponding only to primes in $Q$. A positive integer $n$ is called [i]special[/i] if $p(n)+p(n+1)$ and $q(n)+q(n+1)$ are both even integers. Prove that there is a constant $c>0$ independent of the set $Q$ such that for any positive integer $N>100$, the number of special integers in $[1,N]$ is at least $cN$.
(For example, if $Q=\{3,7\}$, then $p(42)=3$, $q(42)=2$, $p(63)=3$, $q(63)=3$, $p(2022)=3$, $q(2022)=1$.)
Let $p{}$ be an odd prime number. Determine whether there exists a permutation $a_1,\ldots,a_p$ of $1,\ldots,p$ satisfying \[(i-j)a_k+(j-k)a_i+(k-i)a_j\neq 0,\] for all pairwise distinct $i,j,k.$
Define $f: \mathbb{N} \rightarrow \mathbb{N}$ by $$f(n) = \sum \frac{(1+\sum_{i=1}^{n} t_i)!}{(1+t_1) \cdot \prod_{i=1}^{n} (t_i!) }$$
where the sum runs through all $n$-tuples such that $\sum_{j=1}^{n}j \cdot t_j=n$ and $t_j \ge 0$ for all $1 \le j \le n$.
Given a prime $p$ greater than $3$, prove that $$\sum_{1 \le i < j <k \le p-1 } \frac{f(i)}{i \cdot j \cdot k} \equiv \sum_{1 \le i < j <k \le p-1 } \frac{2^i}{i \cdot j \cdot k} \pmod{p}.$$
Let $p_n$ be the $n$-th prime number. Find the remainder when $\Pi_{n=1}^{2549} 2006^{p^2_{n-1}}$ is divided by $13$
The cells of a $1\times 2n$ board are labelled $1,2,...,, n, -n,..., -2, -1$ from left to right. A marker is placed on an arbitrary cell. If the label of the cell is positive, the marker moves to the right a number of cells equal to the value of the label. If the label is negative, the marker moves to the left a number of cells equal to the absolute value of the label. Prove that if the marker can always visit all cells of the board, then $2n + 1$ is prime.
A sequence $x_1, x_2, ..., x_n, ...$ consists of an initial block of $p$ positive distinct integers that then repeat periodically. This means that $\{x_1, x_2, \dots, x_p\}$ are $p$ distinct positive integers and $x_{n+p}=x_n$ for every positive integer $n$. The terms of the sequence are not known and the goal is to find the period $p$. To do this, at each move it possible to reveal the value of a term of the sequence at your choice.
(a) Knowing that $1 \le p \le 10$, find the least $n$ such that there is a strategy which allows to find $p$ revealing at most $n$ terms of the sequence.
(b) Knowing that $p$ is one of the first $k$ prime numbers, find for which values of $k$ there exist a strategy that allows to find $p$ revealing at most $5$ terms of the sequence.
If $ p \geq 5$ is a prime number, then $ 24$ divides $ p^2 \minus{} 1$ without remainder
$ \textbf{(A)}\ \text{never} \qquad
\textbf{(B)}\ \text{sometimes only} \qquad
\textbf{(C)}\ \text{always} \qquad$
$ \textbf{(D)}\ \text{only if } p \equal{}5 \qquad
\textbf{(E)}\ \text{none of these}$
Given natural number n such that, for any natural $a,b$ number $2^a3^b+1$ is not divisible by $n$.Prove that $2^c+3^d$ is not divisible by $n$ for any natural $c$ and $d$
$p(x_1, x_2, ... , x_n)$ is a polynomial with integer coefficients. For each positive integer $r, k(r)$ is the number of $n$-tuples $(a_1, a_2,... , a_n)$ such that $0 \le a_i \le r-1 $ and $p(a_1, a_2, ... , a_n)$ is prime to $r$. Show that if $u$ and $v$ are coprime then $k(u\cdot v) = k(u)\cdot k(v)$, and if p is prime then $k(p^s) = p^{n(s-1)} k(p)$.
Let $p \geq 5$ be a prime.
(a) Show that exists a prime $q \neq p$ such that $q| (p-1)^{p}+1$
(b) Factoring in prime numbers $(p-1)^{p}+1 = \prod_{i=1}^{n}p_{i}^{a_{i}}$ show that:
\[\sum_{i=1}^{n}p_{i}a_{i}\geq \frac{p^{2}}2 \]
Let $ p_{1} <p_{2} <\ldots <p_{24}$ be the prime numbers on the interval $ \left[3,100\right]$. Find the smallest value of $ a\ge 0$ such that $ \sum _{i\equal{}1}^{24}p_{i}^{99!} \equiv a\, \, \left(mod\, 100\right)$.
$\textbf{(A)}\ 24 \qquad\textbf{(B)}\ 25 \qquad\textbf{(C)}\ 48 \qquad\textbf{(D)}\ 50 \qquad\textbf{(E)}\ 99$