Found problems: 1048
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 $n^2 + n + p = 1996$.
Let $ n$ be the largest integer that is the product of exactly $ 3$ distinct prime numbers, $ d$, $ e$, and $ 10d\plus{}e$, where $ d$ and $ e$ are single digits. What is the sum of the digits of $ n$?
$ \textbf{(A)}\ 12 \qquad
\textbf{(B)}\ 15 \qquad
\textbf{(C)}\ 18 \qquad
\textbf{(D)}\ 21 \qquad
\textbf{(E)}\ 24$
Find all triples $(m,p,q)$ such that
\begin{align*}
2^mp^2 +1=q^7,
\end{align*}
where $p$ and $q$ are ptimes and $m$ is a positive integer.
An increasing arithmetic progression consists of one hundred positive integers. Is it possible that every two of them are relatively prime?
Let $p$ and $q$ be distinct odd primes. Show that
$$\bigg\lceil \dfrac{p^q+q^p-pq+1}{pq} \bigg\rceil$$ is even.
Let $p$ be a prime number and $f$ be a bijection from $\left\{0,1,\ldots,p-1\right\}$ to itself. Suppose that for integers $a,b \in \left\{0,1,\ldots,p-1\right\}$, $|f(a) - f(b)|\leqslant 2024$ if $p \mid a^2 - b$. Prove that there exists infinite many $p$ such that there exists such an $f$ and there also exists infinite many $p$ such that there doesn't exist such an $f$.
For a positive integer $k$, let $p(k)$ be the smallest prime that does not divide $k$. Given a positive integer $a$, define the infinite sequence $a_0, a_1, \ldots$ by $a_0 = a$ and, for $n > 0$, $a_n$ is the smallest positive integer with the following properties:
• $a_n$ has not yet appeared in the sequence, that is, $a_n \neq a_i$ for $0 \leq i < n$;
• $(a_{n-1})^{a_n} - 1$ is a multiple of $p(a_{n-1})$.
Prove that every positive integer appears as a term in the sequence, that is, for every positive integer $m$ there is $n$ such that $a_n = m$.
Suppose the prime numbers $p$ and $q$ satisfy $q^2+3p=197p^2+q$.Write $\frac{p}{q}$ as $l+\frac{m}{n}$, where $l,m,n$ are positive integers , $m<n$ and $GCD(m,n)=1$. Find the maximum value of $l+m+n$.
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)
The diagonals of some convex quadrilateral are mutually perpendicular and divide the quadrangle into $4$ triangles, the areas of which are expressed by prime numbers. Prove that a circle can be inscribed in this quadrilateral.
Let $p$ and $q$ be prime numbers. Show that $p^2+q^2+2020$ is composite.
Let $ b, m, n$ be positive integers such that $ b > 1$ and $ m \neq n.$ Prove that if $ b^m \minus{} 1$ and $ b^n \minus{} 1$ have the same prime divisors, then $ b \plus{} 1$ is a power of 2.
A prime $p$ and a positive integer $n$ are given. The product $$(1^3+1)(2^3+1)...((n-1)^3+1)(n^3+1)$$ is divisible by $p^3$. Prove that $p \leq n+1$.
[i]Proposed by Z. Luria[/i]
Find all prime numbers $p$ and $q$ such that $3p^2q+2pq^2=483$
Find all prime numbers $p$ such that $3^p + 5^p -1$ is a prime number.
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]
Find all prime numbers $p$ such that $p^3$ divides the determinant
\[\begin{vmatrix} 2^2 & 1 & 1 & \dots & 1\\1 & 3^2 & 1 & \dots & 1\\ 1 & 1 & 4^2 & & 1\\ \vdots & \vdots & & \ddots & \\1 & 1 & 1 & & (p+7)^2 \end{vmatrix}.\]
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.