Found problems: 721
Does there exist a strictly increasing sequence $\{a_n\}_{n=1}^\infty$ of natural numbers with the following property: for $\forall$ $c\in \mathbb{Z}$ the sequence $c+a_1,c+a_2,...,c+a_n...$ has finite number of primes? Explain your answer.
Find all prime numbers $p$ for which there exist positive integers $x$, $y$, and $z$ such that the number
$x^p + y^p + z^p - x - y - z$
is a product of exactly three distinct prime numbers.
Given distinct prime numbers $p$ and $q$ and a natural number $n \geq 3$, find all $a \in \mathbb{Z}$ such that the polynomial $f(x) = x^n + ax^{n-1} + pq$ can be factored into 2 integral polynomials of degree at least 1.
Find all prime numbers $p$ for which there exist positive integers $x$, $y$, and $z$ such that the number
$x^p + y^p + z^p - x - y - z$
is a product of exactly three distinct prime numbers.
Let us call "absolutely prime" the prime number, if having transposed its digits in an arbitrary order, we obtain prime number again. Prove that its notation cannot contain more than three different digits.
The numbers $2005! + 2, 2005! + 3, ... , 2005! + 2005$ form a sequence of $2004$ consequtive integers, none of which is a prime number.
Does there exist a sequence of $2004$ consequtive integers containing exactly $12$ prime numbers?
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]
$a)$ Let $p$ and $q$ be distinct prime numbers such that $p+q^2$ divides $p^2+q$. Prove that $p+q^2$ divides $pq-1$.
$b)$ Find all prime numbers $p$ such that $p+121$ divides $p^2+11$.
Let $[n]=\{1,2,...,n\}$.Let $p$ be any prime number.
Find how many finite non-empty sets $S\in [p] \times [p]$ are such that $$\displaystyle \large p | \sum_{(x,y) \in S}{x},p | \sum_{(x,y) \in S}{y}$$
$p$ is a prime number such that its remainder divided by 8 is 3. Find all pairs of rational numbers $(x,y)$ that satisfy the following equation.
$$p^2 x^4-6px^2+1=y^2$$
Let $a$ and $b$ be prime numbers such that $a+b = 10000$. Find the sum of the smallest possible value of $a$ and the largest possible value of $a$.
Suppose that $a_1,...,a_{15}$ are prime numbers forming an arithmetic progression with common difference $d > 0$ if $a_1 > 15$ show that $d > 30000$
Let $p$ be a prime number. We construct a directed graph of $p$ vertices, labeled with integers from $0$ to $p-1$. There is an edge from vertex $x$ to vertex $y$ if and only if $x^2+1\equiv y \pmod{p}$. Let $f(p)$ denotes the length of the longest directed cycle in this graph. Prove that $f(p)$ can attain arbitrarily large values.
a) Let $f,g:\mathbb{Z}\rightarrow\mathbb{Z}$ be one to one maps. Show that the function $h:\mathbb{Z}\rightarrow\mathbb{Z}$ defined by $h(x)=f(x)g(x)$, for all $x\in\mathbb{Z}$, cannot be a surjective function.
b) Let $f:\mathbb{Z}\rightarrow\mathbb{Z}$ be a surjective function. Show that there exist surjective functions $g,h:\mathbb{Z}\rightarrow\mathbb{Z}$ such that $f(x)=g(x)h(x)$, for all $x\in\mathbb{Z}$.
Find all prime numbers $p$, for which there exist $x, y \in \mathbb{Q}^+$ and $n \in \mathbb{N}$, satisfying
$x+y+\frac{p}{x}+\frac{p}{y}=3n$.
Let $p \equiv 2 \pmod 3$ be a prime, $k$ a positive integer and $P(x) = 3x^{\frac{2p-1}{3}}+3x^{\frac{p+1}{3}}+x+1$. For any integer $n$, let $R(n)$ denote the remainder when $n$ is divided by $p$ and let $S = \{0,1,\cdots,p-1\}$. At each step, you can either (a) replaced every element $i$ of $S$ with $R(P(i))$ or (b) replaced every element $i$ of $S$ with $R(i^k)$. Determine all $k$ such that there exists a finite sequence of steps that reduces $S$ to $\{0\}$.
[i]Proposed by fattypiggy123[/i]
Three members of the Euclid Middle School girls' softball team had the following conversation.
Ashley: I just realized that our uniform numbers are all $2$-digit primes.
Bethany: And the sum of your two uniform numbers is the date of my birthday earlier this month.
Caitlin: That's funny. The sum of your two uniform numbers is the date of my birthday later this month.
Ashley: And the sum of you two uniform numbers is today's date.
What number does Caitlin wear?
$\textbf{(A) }11\qquad\textbf{(B) }13\qquad\textbf{(C) }17\qquad\textbf{(D) }19\qquad \textbf{(E) }23$
Prove that there exists a prime number $p$, such that the sum of digits of $p$ is a composite odd integer. Find the smallest such $p$.
The ratio of prime numbers $p$ and $q$ does not exceed 2 ($p\ne q$).
Prove that there are two consecutive positive integers such that
the largest prime divisor of one of them is $p$ and that of the other is $q$.
Let $p \ge 5$ be a prime number. For a positive integer $k$, let $R(k)$ be the remainder when $k$ is divided by $p$, with $0 \le R(k) \le p-1$. Determine all positive integers $a < p$ such that, for every $m = 1, 2, \cdots, p-1$, $$ m + R(ma) > a. $$
Find all prime numbers of the form $\tfrac{1}{11} \cdot \underbrace{11\ldots 1}_{2n \textrm{ ones}}$, where $n$ is a natural number.
Find all positive $k$ such that product of the first $k$ odd prime numbers, reduced by 1 is exactly degree of natural number (which more than one).
Let $ p,q,r$ be 3 prime numbers such that $ 5\leq p <q<r$. Knowing that $ 2p^2\minus{}r^2 \geq 49$ and $ 2q^2\minus{}r^2\leq 193$, find $ p,q,r$.
Let $p$ be a prime number, determine all positive integers $(x, y, z)$ such that:
$x^p + y^p = p^z$
Find all pairs of primes $(p, q)$ for which $p-q$ and $pq-q$ are both perfect squares.