Found problems: 1048
The $2011$th prime number is $17483$ and the next prime is $17489$.
Does there exist a sequence of $2011^{2011}$ consecutive positive integers that contain exactly $2011$ prime numbers?
Let $p,q$ be positive prime numbers and suppose $q>5$. Prove that if $q \mid 2^{p}+3^{p}$, then $q>p$.
[i]Laurentiu Panaitopol[/i]
Find all prime numbers $p$ and $q$ such that $pq-p-q+3$ is a perfect square.
For any integer $d > 0,$ let $f(d)$ be the smallest possible integer that has exactly $d$ positive divisors (so for example we have $f(1)=1, f(5)=16,$ and $f(6)=12$). Prove that for every integer $k \geq 0$ the number $f\left(2^k\right)$ divides $f\left(2^{k+1}\right).$
[i]Proposed by Suhaimi Ramly, Malaysia[/i]
A prime number has the property that however its decimal digits are permuted, the obtained number is also prime. Prove that this number has at most three different digits. Also prove a stronger statement.
Let $p=ab+bc+ac$ be a prime number where $a,b,c$ are different two by two, show that $a^3,b^3,c^3$ gives different residues modulo $p$
Let $n$ be a positive integer and $p$ be a prime number of the form $8k+5$. A polynomial $Q$ of degree at most $2023$ and nonnegative integer coefficients less than or equal to $n$ will be called "cool" if
\[p\mid Q(2)\cdot Q(3) \cdot \ldots \cdot Q(p-2)-1.\]
Prove that the number of cool polynomials is even.
Find all prime number pairs $(p,q)$ such that $p(p^2-p-1)=q(2q+3).$
Find all triples $(x; y; p)$ of two non-negative integers $x, y$ and a prime number p such that $ p^x-y^p=1 $
Prove that there exists a sequence of unbounded positive integers $a_1\leq a_2\leq a_3\leq\cdots$, such that there exists a positive integer $M$ with the following property: for any integer $n\geq M$, if $n+1$ is not prime, then any prime divisor of $n!+1$ is greater than $n+a_n$.
We observe that number $10001=73\cdot137$ is not prime. Show that every member of infinite sequence $10001, 100010001, 1000100010001,...$ is not prime
Given a positive prime number $p$. Prove that there exist a positive integer $\alpha$ such that $p|\alpha(\alpha-1)+3$, if and only if there exist a positive integer $\beta$ such that $p|\beta(\beta-1)+25$.
Let $A$ be a set of $2025$ non-negative integers and $f: \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$ be a function with the following two properties:
1) For every two distinct positive integers $x,y$ there exists $a\in A$, such that $x-y$ divides $f(x+a) - f(y+a)$.
2) For every positive integer $N$ there exists a positive integer $t$ such that $f(x) \neq f(y)$ whenever $x,y \in [t, t+N]$ are distinct.
Prove that there are infinitely many primes $p$ such that $p$ divides $f(x)$ for some positive integer $x$.
Let $a,b$ be integers and $p$ be a prime number such that:
(i) $p$ is the greatest common divisor of $a$ and $b$;
(ii) $p^2$ divides $a$.
Prove that the polynomial $x^{n+2}+ax^{n+1}+bx^{n}+a+b$ cannot be decomposed into the product of two polynomials with integer coefficients and degree greater than $1$.
Prove that if the product $1\cdot 2\cdot ...\cdot n$ ($n> 3$) is not divisible by $n + 1$, then $n + 1$ is prime.
Are there prime $p$ and $q$ larger than $3$, such that $p^2-1$ is divisible by $q$ and $q^2-1$ divided by $p$?
Find the largest prime number $p$ such that when $2012!$ is written in base $p$, it has at least $p$ trailing zeroes.
[i]Author: Alex Zhu[/i]
Prove that if $ a $ is an integer different from $ 1 $ and $ - 1 $, then $ a^4 + 4 $ is not a prime number.
The following sequence of positive integers $a_1, a_2, ..., a_{400}$ satisfies the relationship $a_{n+1} = \tau (a_n) + \tau (n)$ for all $1 \le n \le 399$, where $\tau (k) $ is the number of positive integer divisors that $k$ has. Prove that in the sequence there are no more than $210$ prime numbers.
Let $S_n$ denote the sum of the first $n$ prime numbers. Prove that for any $n$ there exists the square of an integer between $S_n$ and $S_{n+1}$.
For all positive integers $a$ and $b$, we de ne $a @ b = \frac{a - b}{gcd(a, b)}$ .
Show that for every integer $n > 1$, the following holds:
$n$ is a prime power if and only if for all positive integers $m$ such that $m < n$, it holds that $gcd(n, n @m) = 1$.
Let $x$ be an integer of $n$ digits, all equal to $ 1$. Show that if $x$ is prime, then $n$ is also prime.
Four integers are marked on a circle. On each step we simultaneously replace each number by the difference between this number and next number on the circle, moving in a clockwise direction; that is, the numbers $ a,b,c,d$ are replaced by $ a\minus{}b,b\minus{}c,c\minus{}d,d\minus{}a.$ Is it possible after 1996 such to have numbers $ a,b,c,d$ such the numbers $ |bc\minus{}ad|, |ac \minus{} bd|, |ab \minus{} cd|$ are primes?
Let $p$ be a prime number. Troy and Abed are playing a game. Troy writes a positive integer $X$ on the board, and gives a sequence $(a_n)_{n\in\mathbb{N}}$ of positive integers to Abed. Abed now makes a sequence of moves. The $n$-th move is the following:
$$\text{ Replace } Y \text{ currently written on the board with either } Y + a_n \text{ or } Y \cdot a_n.$$
Abed wins if at some point the number on the board is a multiple of $p$. Determine whether Abed can win, regardless of Troy’s choices, if
$a) p = 10^9 + 7$;
$b) p = 10^9 + 9$.
[i]Remark[/i]: Both $10^9 + 7$ and $10^9 + 9$ are prime.
[i]Proposed by Ivan Novak[/i]
Exists a positive integer $n$ such that the number $\underbrace{1...1}_{n \,ones} 2 \underbrace{1...1}_{n \, ones}$ is a prime number?