Found problems: 1048
Prove that there are no $1999$ primes in an arithmetic progression that are all less than $12345$.
There exists a block of 1000 consecutive positive integers containing no prime numbers, namely, $1001!+2,1001!+3,...,1001!+1001$. Does there exist a block of 1000 consecutive positive intgers containing exactly five prime numbers?
Find all triples $(p, x, y)$ consisting of a prime number $p$ and two positive integers $x$ and $y$ such that $x^{p -1} + y$ and $x + y^ {p -1}$ are both powers of $p$.
[i]Proposed by Belgium[/i]
Prove that if $p$ and $q$ are two prime numbers, such that
$$p+p^2+p^3+...+p^q=q+q^2+q^3+...+q^p,$$
then $p=q$.
Determine the smallest prime $p$ such that $2018!$ is divisible by $p^{3}$ , but not divisible by $p^{4}$.
Among the positive integers less than $100$, each of whose digits is a prime number, one is selected at random. What is the probablility that the selected number is prime?
$\textbf{(A) } \dfrac{8}{99}
\qquad\textbf{(B) } \dfrac{2}{5}
\qquad\textbf{(C) } \dfrac{9}{20}
\qquad\textbf{(D) } \dfrac{1}{2}
\qquad\textbf{(E) } \dfrac{9}{16}
$
Decide whether there are infinitely many primes $p$ having a multiple in the form $n^2 + n + 1$ for some natural number $n$
Let $(a_n)_{n = 0}^{\infty} $ be a sequence of positive integers such that for every $n \geq 0$ it is true that
$$a_{n+2} = a_0 a_1 + a_1 a_2 + ... + a_n a_{n+1} - 1 $$
a) Prove that there exist a prime number which divides infinitely many $a_n$
b) Prove that there exist infinitely many such prime numbers
Let $k$ be a positive integer and let $S$ be a finite set of odd prime numbers. Prove that there is at most one way (up to rotation and reflection) to place the elements of $S$ around the circle such that the product of any two neighbors is of the form $x^2+x+k$ for some positive integer $x$.
For arbitrary positive integers $a, b$, denote $a @ b =\frac{a-b}{gcd(a,b)}$
Let $n$ be a positive integer. Prove that the following conditions are equivalent:
(i) $gcd(n, n @ m) = 1$ for every positive integer $m < n$,
(ii) $n = p^k$ where $p$ is a prime number and $k$ is a non-negative integer.
Let's call any natural number "very prime" if any number of consecutive digits (in particular, a digit or number itself) is a prime number. For example, $23$ and $37$ are "very prime" numbers, but $237$ and $357$ are not. Find the largest "prime" number (with justification!).
Let $n\geq3$ be an integer. We say that an arrangement of the numbers $1$, $2$, $\dots$, $n^2$ in a $n \times n$ table is [i]row-valid[/i] if the numbers in each row can be permuted to form an arithmetic progression, and [i]column-valid[/i] if the numbers in each column can be permuted to form an arithmetic progression. For what values of $n$ is it possible to transform any row-valid arrangement into a column-valid arrangement by permuting the numbers in each row?
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]
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$
Show that the cube roots of three distinct prime numbers cannot be three terms (not necessarily consecutive) of an arithmetic progression.
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?
Find all pairs $(p,q)$ of prime numbers such that
$$ p(p^2 - p - 1) = q(2q + 3) .$$
Let $P(x)$ be a polynomial with integer coefficients. We will denote the set of all prime numbers by $\mathbb P$. Show that the set $\mathbb S := \{p\in\mathbb P : \exists\text{ }n \text{ s.t. }p\mid P(n)\}$ is finite if and only if $P(x)$ is a non-zero constant polynomial.
For a given odd prime number $p$, define $f(n)$ the remainder of $d$ divided by $p$, where $d$ is the biggest divisor of $n$ which is not a multiple of $p$. For example when $p=5$, $f(6)=1, f(35)=2, f(75)=3$. Define the sequence $a_1, a_2, \ldots, a_n, \ldots$ of integers as the followings:
[list]
[*]$a_1=1$
[*]$a_{n+1}=a_n+(-1)^{f(n)+1}$ for all positive integers $n$.
[/list]
Determine all integers $m$, such that there exist infinitely many positive integers $k$ such that $m=a_k$.
Given a finite set $ P$ of prime numbers, prove that there exists a positive integer $ x$ such that it can be written in the form $ a^p \plus{} b^p$ ($ a,b$ are positive integers), for each $ p\in P$, and cannot be written in that form for each $ p$ not in $ P$.
The positive integers $x_1, \cdots , x_n$, $n \geq 3$, satisfy $x_1 < x_2 <\cdots< x_n < 2x_1$. Set $P = x_1x_2 \cdots x_n.$ Prove that if $p$ is a prime number, $k$ a positive integer, and $P$ is divisible by $pk$, then $\frac{P}{p^k} \geq n!.$
Let $ a $ , $ b $ , $ c $ , $ d $ , $ (a + b + c + 18 + d) $ , $ (a + b + c + 18 - d) $ , $ (b + c) $ , and $ (c + d) $ be distinct prime numbers such that $ a + b + c = 2010 $, $ a $, $ b $, $ c $, $ d \neq 3 $ , and $ d \le 50 $. Find the maximum value of the difference between two of these prime numbers.
Let $P (n)$ and $Q (n)$ be two polynomials (not constant) whose coefficients are integers not negative. For each positive integer $n$, define $x_n = 2016^{P (n)} + Q (n)$. Prove that there exist infinite primes $p$ for which there is a positive integer $m$, squarefree, such that $p | x_m$.
Clarification: A positive integer is squarefree if it is not divisible by the square of any prime number.
Define a number to be $boring$ if all the digits of the number are the same. How many positive integers less than $10000$ are both prime and boring?
Prove that there are infinitely many prime numbers that divide at least one integer of the form $2^{n^3+1}-3^{n^2+1}+5^{n+1}$ where $n$ is a positive integer.