Found problems: 1048
Find all positive integers n such that $n^5 + n + 1$ is prime.
Prove that for any prime ${p}$, different than ${2}$ and ${5}$, there exists such a multiple of ${p}$ whose digits are all nines. For example, if ${p = 13}$, such a multiple is ${999999 = 13 * 76923}$.
Determine all pairs $(p, q)$ of prime numbers such that $p^p + q^q + 1$ is divisible by $pq.$
Prove that there exist infinitely many positive integers $n$ such that the greatest prime divisor of $n^2+1$ is less than $n \cdot \pi^{-2019}.$
Famous French mathematician Pierre Fermat believed that all numbers of the form $F_n = 2^{2^n} + 1$ are prime for all non-negative integers $n$. Indeed, one can check that $F_0 = 3$, $F_1 = 5$, $F_2 = 17$, $F_3 = 257$ are all prime.
a) Prove that $F_5$ is divisible by $641$. (Hence Fermat was wrong.)
b) Prove that if $k \ne n$ then $F_k$ and $F_n$ are relatively prime (i.e. they do not have any common divisor except $1$)
(Notice: using b) one can prove that there are infinitely many prime numbers)
Find all primes that can be written both as a sum and as a difference of two primes (note that $ 1$ is not a prime).
Consider the statement, "If $n$ is not prime, then $n-2$ is prime." Which of the following values of $n$ is a counterexample to this statement?
$\textbf{(A) } 11 \qquad \textbf{(B) } 15 \qquad \textbf{(C) } 19 \qquad \textbf{(D) } 21 \qquad \textbf{(E) } 27$
For each positive integer $ n$, let $ f(n)\equal{}n^4\minus{}360n^2\plus{}400$. What is the sum of all values of $ f(n)$ that are prime numbers?
$ \textbf{(A)}\ 794\qquad
\textbf{(B)}\ 796\qquad
\textbf{(C)}\ 798\qquad
\textbf{(D)}\ 800\qquad
\textbf{(E)}\ 802$
Let $p$ be a prime number and $F=\left \{0,1,2,...,p-1 \right \}$. Let $A$ be a proper subset of $F$ that satisfies the following property: if $a,b \in A$, then $ab+1$ (mod $p$) $ \in A$.
How many elements can $A$ have? (Justify your answer.)
Let $m$ and $n$ be natural numbers such that $m^3-n^3$ is a prime number. What is the remainder of the number $m^3-n^3$ when divided by $6$?
Find the number of positive integers $n$ with exactly $1974$ factors such that no prime greater than $40$ divides $n$, and $n$ ends in one of the digits $1$, $3$, $7$, $9$. (Note that $1974 = 2 \cdot 3 \cdot 7 \cdot 47$.)
[i]Proposed by Yonah Borns-Weil[/i]
Let $ p$ be an odd prime. Determine positive integers $ x$ and $ y$ for which $ x \leq y$ and $ \sqrt{2p} \minus{} \sqrt{x} \minus{} \sqrt{y}$ is non-negative and as small as possible.
Find all natural numbers $ n$ for which every natural number whose decimal representation has $ n \minus{} 1$ digits $ 1$ and one digit $ 7$ is prime.
Let $p$ be an odd prime. Find all positive integers $n$ for which $\sqrt{n^2-np}$ is a positive integer.
Find all possible values of $n \ge 1$ for which there exist $n$ consecutive positive integers whose sum is a prime number.
Let $p>5$ be a prime number. Show that $p-4$ cannot be the fourth power of a prime number.
Solve in prime numbers:
$ \{\begin{array}{c}\ \ 2a - b + 7c = 1826 \
3a + 5b + 7c = 2007 \end{array}$
For each positive integer $m> 1$, let $P (m)$ be the product of all prime numbers that divide $m$.
Define the sequence $a_1, a_2, a_3,...$ as followed:
$a_1> 1$ is an arbitrary positive integer,
$a_{n + 1} = a_n + P (a_n)$ for each positive integer $n$.
Prove that there exist positive integers $j$ and $k$ such that $a_j$ is the product of the first $k$ prime numbers.
Find all pairs of positive integers $(x, y)$ such that $\frac{xy^3}{x+y}$ is the cube of a prime.
Find all the pairs of positive integers $(x,p)$ such that p is a prime, $x \leq 2p$ and $x^{p-1}$ is a divisor of $ (p-1)^{x}+1$.
$n$ is natural number and $p$ is prime number. If $1+np$ is square of natural number then prove that $n+1$ is equal to some sum of $p$ square of natural numbers.
It is given that the number $4^{11}+1$ is divisible by some prime greater than $1000$. Determine this prime.
[i] Proposed by David Altizio [/i]
A positive integer is called [i]tico[/i] if it is the product of three different prime numbers that add up to 74. Verify that 2014 is tico. Which year will be the next tico year? Which one will be the last tico year in history?
Show that when a prime number is divided by $30$, the remainder is $1$ or a prime number. Shows that if it is divided by $60$ or $90$ the same thing does not happen.
Find all positive integers $n>2$ such that
$$ n! \mid \prod_{ p<q\le n, p,q \, \text{primes}} (p+q)$$