Found problems: 721
Find the greatest possible value of $pq + r$, where p, q, and r are (not necessarily distinct) prime numbers satisfying $pq + qr + rp = 2016$.
Let $p$ be a prime number. Find all polynomials $P$ with integer coefficients with the following properties:
$(a)$ $P(x)>x$ for all positive integers $x$.
$(b)$ The sequence defined by $p_0:=p$, $p_{n+1}:=P(p_n)$ for all positive integers $n$, satisfies the property that for all positive integers $m$ there exists some $l\geq 0$ such that $m\mid p_l$.
Let $p$ be a prime number.
Prove that it is possible to choose a permutation $a_1, a_2,...,a_p$ of $1,2,...,p$ such that the numbers $a_1, a_1a_2, a_1a_2a_3,..., a_1a_2a_3...a_p$ all have different remainder upon division by $p$.
Let $p$ and $q$ be different prime numbers. Solve the following system in integers:
\[\frac{z+ p}x+\frac{z-p}y= q,\\ \frac{z+ p}y -\frac{z-p}x= q.\]
You are given a sequence of $58$ terms; each term has the form $P+n$ where $P$ stands for the product $2 \times 3 \times 5 \times... \times 61$ of all prime numbers less than or equal to $61$, and $n$ takes, successively, the values $2, 3, 4, ...., 59$. let $N$ be the number of primes appearing in this sequence. Then $N$ is:
$ \textbf{(A)}\ 0\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 17\qquad\textbf{(D)}\ 57\qquad\textbf{(E)}\ 58 $
Find all prime numbers $p$ for which $x^4\equiv -1\, (mod\, p)$ has a solution.
Do there exist
(a) four
(b) five
distinct positive integers such that the sum of any three of them is a prime number?
(V Senderov)
Let $n \geq 3$ be a positive integers and $p$ be a prime number such that $p > 6^{n-1} - 2^n + 1$. Let $S$ be the set of $n$ positive integers with different residues modulo $p$. Show that there exists a positive integer $c$ such that there are exactly two ordered triples $(x,y,z) \in S^3$ with distinct elements, such that $x-y+z-c$ is divisible by $p$.
The Fibonacci sequence $F_n$ is defined by $F_0=0,F_1=1$ and the recurrence relation $F_n=F_{n-1}+F_{n-2}$ for all integers $n\geq2$. Let $p\geq3$ be a prime number.
[list=a]
[*] Prove that $F_{p-1}+F_{p+1}-1$ is divisible by $p$.
[*] Prove that $F_{p^{k+1}-1}+F_{p^{k+1}+1}-\left(F_{p^k-1}+F_{p^k+1}\right)$ is divisible by $p^{k+1}$ for any positive integer $k$.
[/list]
Prove that $2005^2$ can be written in at least $4$ ways as the sum of 2 perfect (non-zero) squares.
Prove that for any prime number $p > 3$ exist integers $x, y, k$ that meet conditions: $0 < 2k < p$ and $kp + 3 = x^2 + y^2$.
For how many three-element sets of positive integers $\{a,b,c\}$ is it true that $a \times b \times c = 2310$?
$\textbf{(A)}\ 32 \qquad
\textbf{(B)}\ 36 \qquad
\textbf{(C)}\ 40 \qquad
\textbf{(D)}\ 43 \qquad
\textbf{(E)}\ 45$
Prove that the arithmetic progression $3,7,11,15,...$. contains infinitely many prime numbers.
Find the sum of all prime numbers $p$ which satisfy \[p = a^4 + b^4 + c^4 - 3\] for some primes (not necessarily distinct) $a$, $b$ and $c$.
Find the smallest prime number that can not be written in the form $\left| 2^a-3^b \right|$ with non-negative integers $a,b$.
Let $a,b,c$ and $d$ be prime numbers such that $a>3b>6c>12d$ and $a^2-b^2+c^2-d^2=1749$. Determine all possible values of $a^2+b^2+c^2+d^2$ .
Suppose that for some $m,n\in\mathbb{N}$ we have $\varphi (5^m-1)=5^n-1$, where $\varphi$ denotes the Euler function. Show that $(m,n)>1$.
Let's call a positive integer $n$ special, if there exist two nonnegativ integers ($a, b$), such that $n=2^a\times 3^b$.
Prove that if $k$ is a positive integer, then there are at most two special numbers greater then $k^2$ and less than $k^2+2k+1$.
Let $S$ be a finite set of integers, each greater than $1$. Suppose that for each integer $n$ there is some $s\in S$ such that $\gcd(s,n)=1$ or $\gcd(s,n)=s$. Show that there exist $s,t\in S$ such that $\gcd(s,t)$ is prime.
[b]4.[/b] Find all positive integers $\alpha , \beta (\alpha >1)$ and all prime numbers $p, q, r$ which satisfy the equation $p^{\alpha}= q^{\beta}+r^{\alpha}$ ($\alpha , \beta , p, q, r$ need not necessarily be different). [b](N. 12)[/b]
How many subsets of the set $\{1, 2, 3, \ldots, 12\}$ contain exactly one or two prime numbers?
If the representation of a positive number as a product of powers of distinct prime numbers contains no even powers other than $0$s, we will call the number singular. At most how many consequtive singular numbers are there?
$ \textbf{(A)}\ 6 \qquad \textbf{(B)}\ 7 \qquad \textbf{(C)}\ 8 \qquad \textbf{(D)}\ 9 \qquad \textbf{(E)}\ \text{None}$
Let $p,q$ be prime numbers such that their sum isn't divisible by $3$. Find the all $(p,q,r,n)$ positive integer quadruples satisfy:
$$p+q=r(p-q)^n$$
[i]Proposed by Şahin Emrah[/i]
Let $S_n$ denote the set of permutations of the sequence $(1,2,\dots, n)$. For every permutation $\pi=(\pi_1, \dots, \pi_n)\in S_n$, let $\mathrm{inv}(\pi)$ be the number of pairs $1\le i < j \le n$ with $\pi_i>\pi_j$; i. e. the number of inversions in $\pi$. Denote by $f(n)$ the number of permutations $\pi\in S_n$ for which $\mathrm{inv}(\pi)$ is divisible by $n+1$.
Prove that there exist infinitely many primes $p$ such that $f(p-1)>\frac{(p-1)!}{p}$, and infinitely many primes $p$ such that $f(p-1)<\frac{(p-1)!}{p}$.
(Proposed by Fedor Petrov, St. Petersburg State University)
a) Show that it is possible to pair off the numbers $1,2,3,\ldots ,10$ so that the sums of each of the five pairs are five different prime numbers.
b) Is it possible to pair off the numbers $1,2,3,\ldots ,20$ so that the sums of each of the ten pairs are ten different prime numbers?