Found problems: 364
Find the smallest $n$ such that $n^2 -n+11$ is the product of four primes (not necessarily distinct).
Let $p$ be a prime number. Determine the largest possible $n$ such that the following holds: it is possible to fill an $n\times n$ table with integers $a_{ik}$ in the $i$th row and $k$th column, for $1\le i,k\le n$, such that for any quadruple $i,j,k,l$ with $1\le i<j\le n$ and $1\le k<l\le n$, the number $a_{ik}a_{jl}-a_{il}a_{jk}$ is not divisible by $p$.
[i]Proposed by oneplusone[/i]
Find all positive integers $n$ such that $n=q(q^2-q-1)=r(2r+1)$ for some primes $q$ and $r$.
B.Gilevich
Suppose that $p,q$ are prime numbers such that $\sqrt{p^2 +7pq+q^2}+\sqrt{p^2 +14pq+q^2}$ is an integer.
Show that $p = q$.
Prove that for all prime $p \ge 5$, there exist an odd prime $q \not= p$ such that $q$ divides $(p-1)^p + 1$
Let $ \left( s_n \right)_{n\ge 1 } $ be a sequence with $ s_1 $ and defined recursively as $ s_{n+1}=s_n^2-s_n+1. $
Prove that any two terms of this sequence are coprime.
[i]Dan Nedeianu[/i]
Prove that if $n$ is a natural number such that $1 + 2^n + 4^n$ is prime then $n = 3^k$ for some $k \in N_0$.
Determine whether exist a prime number $p$ and natural number $n$ such that $n^2 + n + p = 1996$.
For a positive integer $n$, let $p(n)$ denote the largest prime divisor of $n$. Show that there exist infinitely many positive integers m such that $p(m-1) < p(m) < p(m + 1)$.
Do there exist (not necessarily distinct) primes $p_1,..., p_k$ and $q_1,...,q_n$ such that $$p_1! \cdot \cdot \cdot p_k! \cdot 2017 = q_1! \cdot \cdot \cdot q_n! \cdot 2016 \,\,?$$
(Paolo Leonetti)
We say positive integer $n$ is $\emph{metallic}$ if there is no prime of the form $m^2-n$. What is the sum of the three smallest metallic integers?
[i] Proposed by Lewis Chen [/i]
The natural numbers from $1$ to $50$ are written down on the blackboard. At least how many of them should be deleted, in order that the sum of any two of the remaining numbers is not a prime?
Find all positive integers $n>2$ such that
$$ n! \mid \prod_{ p<q\le n, p,q \, \text{primes}} (p+q)$$
Let $a\geq 2$ be an integer. Prove that there exists a positive integer $b$ with the following property: For each positive integer $n$, there is a prime number $p$ (possibly depending on $a,b,n$) such that $a^n + b$ is divisible by $p$, but not divisible by $p^2$.
Consider the prime numbers $n_1< n_2 <...< n_{31}$. Prove that if $30$ divides $n_1^4 + n_2^4+...+n_{31}^4$, then among these numbers one can find three consecutive primes.
a) Suppose that $n$ is an odd integer. Prove that $k(n-k)$ is divisible by $2$ for all positive integers $k$.
b) Find an integer $k$ such that $k(100-k)$ is not divisible by $11$.
c) Suppose that $p$ is an odd prime, and $n$ is an integer.
Prove that there is an integer $k$ such that $k(n-k)$ is not divisible by $p$.
d) Suppose that $p,q$ are two different odd primes, and $n$ is an integer.
Prove that there is an integer $k$ such that $k(n-k)$ is not divisible by any of $p,q$.
The radius $r$ of a circle with center at the origin is an odd integer.
There is a point ($p^m, q^n$) on the circle, with $p,q$ prime numbers and $m,n$ positive integers.
Determine $r$.
Find all prime numbers $p$ for which $\frac{2^{p-1} -1}{p}$ is a perfect square.
Let $n$ be a natural number, $n > 2$. Prove that if $\frac{b^n-1}{b-1}$ is a prime power for some positive integer $b$ then $n$ is prime.
Prove that if a triangle has integral side lengths and its circumradius is a prime number then the triangle is right-angled.
There are $n$ children in a room. Each child has at least one piece of candy. In Round $1$, Round $2$, etc., additional pieces of candy are distributed among the children according to the following rule:
In Round $k$, each child whose number of pieces of candy is relatively prime to $k$ receives an additional piece.
Show that after a sufficient number of rounds the children in the room have at most two different numbers of pieces of candy.
[i](Proposed by Theresia Eisenkölbl)[/i]
The two pairs of consecutive natural numbers $(8, 9)$ and $(288, 289)$ have the following property: in each pair, each number contains each of its prime factors to a power not less than $2$. Prove that there are infinitely many such pairs.
(A Andjans, Riga)
Find all prime numbers $p$ such that $5^p + 12^p$ is a perfect square
Let $p>2$ be a prime and $a,b$ positive integers such that \[\frac ab=1+\frac12+\frac13+\cdots+\frac{1}{p-1}.\] Show that $p$ is a divisor of $a.$
Let $p_1,p_2,p_3,p_4$ be four distinct primes. Prove that there is no polynomial $Q(x) = ax^3 + bx^2 + cx + d$ with integer coefficients such that $|Q(p_1)| =|Q(p_2)| = |Q(p_3)|= |Q(p_4 )| = 3$.