Found problems: 721
is there a function $f:\mathbb{N}\rightarrow \mathbb{N}$ such that
$i) \exists n\in \mathbb{N}:f(n)\neq n$
$ii)$ the number of divisors of $m$ is $f(n)$ if and only if the number of divisors of $f(m)$ is $n$
Let $p$ and $q$ be prime numbers such that $p^2+pq+q^2$ is perfect square. Prove that $p^2-pq+q^2$ is prime
Assume $\Omega(n),\omega(n)$ be the biggest and smallest prime factors of $n$ respectively . Alireza and Amin decided to play a game. First Alireza chooses $1400$ polynomials with integer coefficients. Now Amin chooses $700$ of them, the set of polynomials of Alireza and Amin are $B,A$ respectively . Amin wins if for all $n$ we have :
$$\max_{P \in A}(\Omega(P(n))) \ge \min_{P \in B}(\omega(P(n)))$$
Who has the winning strategy.
Proposed by [i]Alireza Haghi[/i]
Prove that there exist two natural numbers $a,b$ such that $|a-m|+|b-n|>1000$ for any relatively prime natural numbers $m,n$.
How many subsets of $\{2,3,4,5,6,7,8,9\}$ contain at least one prime number?
$\textbf{(A)} \text{ 128} \qquad \textbf{(B)} \text{ 192} \qquad \textbf{(C)} \text{ 224} \qquad \textbf{(D)} \text{ 240} \qquad \textbf{(E)} \text{ 256}$
Solve the equation for primes $p$ and $q$: $$p^3-q^3=pq^3-1.$$
Palmer correctly computes the product of the first $1,001$ prime numbers. Which of the following is NOT a factor of Palmer's product?
$\text{(A) }2,002\qquad\text{(B) }3,003\qquad\text{(C) }5,005\qquad\text{(D) }6,006\qquad\text{(E) }7,007$
Call a natural number [i]simple[/i] if it is not divisible by any square of a prime number (in other words it is square-free).
Prove that there are infinitely many positive integers $n$ such that both $n$ and $n+1$ are [i]simple[/i].
For a prime number $p$, there exists $n\in\mathbb{Z}_+$, $\sqrt{p+n}+\sqrt{n}$ is an integer, then
$\text{(A)}$ there is no such $p$
$\text{(B)}$ there in only one such $p$
$\text{(C)}$ there is more than one such $p$, but finitely many
$\text{(D)}$ there are infinitely many such $p$
For positive integers $m, n$ define the function $f_n(m) = 1^{2n} + 2^{2n} + 3^{2n} + \ldots +m^{2n}$. Prove that there are only finitely many pairs of positive integers $(a, b)$ such that $f_n(a) + f_n(b)$ is a prime number.
[i]Proposed by Nazar Serdyuk[/i]
There is a prime number $p$ such that $16p+1$ is the cube of a positive integer. Find $p$.
Find all prime numbers $p,q,r,k$ such that $pq+qr+rp = 12k+1$
Let $p > 10^9$ be a prime number such that $4p + 1$ is also prime.
Prove that the decimal expansion of $\frac{1}{4p+1}$ contains all the digits $0,1, \ldots, 9$.
Find all prime numbers $p,q$ such that:
$$p^4+p^3+p^2+p=q^2+q$$
A point $ P$ of integer coordinates in the Cartesian plane is said [i]visible[/i] if the segment $ OP$ does not contain any other points with integer coordinates (except its ends). Prove that for any $ n\in\mathbb{N}^*$ there exists a visible point $ P_{n}$, at distance larger than $ n$ from any other visible point.
[b]Dan Schwarz[/b]
Prove that there exists no in nite sequence of prime numbers $p_0, p_1, p_2,...$ such that for all positive integers $k$:
$p_k = 2p_{k-1} + 1$ or $p_k = 2p_{k-1} - 1$.
Find all integer triples $(p,m,n)$ that satisfy:
$p^m-n^3=27$ where $p$ is a prime number.
Alex and Bruno play the following game: each one, in your turn, the player writes, exactly one digit, in the right of the last number written. The game finishes if we have a number with $6$ digits( distincts ) and Alex starts the game. Bruno wins if the number with $6$ digits is a prime number, otherwise Alex wins.
Which player has the winning strategy?
A positive proper divisor is a positive divisor of a number, excluding itself. For positive integers $n \ge 2$, let $f(n)$ denote the number that is one more than the largest proper divisor of $n$. Determine all positive integers $n$ such that $f(f(n)) = 2$.
Consider a prime number $p\geqslant 11$. We call a triple $a,b,c$ of natural numbers [i]suitable[/i] if they give non-zero, pairwise distinct residues modulo $p{}$. Further, for any natural numbers $a,b,c,k$ we define \[f_k(a,b,c)=a(b-c)^{p-k}+b(c-a)^{p-k}+c(a-b)^{p-k}.\]Prove that there exist suitable $a,b,c$ for which $p\mid f_2(a,b,c)$. Furthermore, for each such triple, prove that there exists $k\geqslant 3$ for which $p\nmid f_k(a,b,c)$ and determine the minimal $k{}$ with this property.
[i]Călin Popescu and Marian Andronache[/i]
For which $ p$ prime numbers, there is an integer root of the polynominal $ 1 \plus{} p \plus{} Q(x^1)\cdot\ Q(x^2)\ldots\ Q(x^{2p \minus{} 2})$ such that $ Q(x)$ is a polynominal with integer coefficients?
Let $p_1<p_2<p_3<p_4$ and $q_1<q_2<q_3<q_4$ be two sets of prime numbers, such that $p_4 - p_1 = 8$ and $q_4 - q_1= 8$. Suppose $p_1 > 5$ and $q_1>5$. Prove that $30$ divides $p_1 - q_1$.
For every integer $n\geq 3$, let $s_n$ be the sum of all primes (strictly) less than $n$. Are there infinitely many integers $n\geq 3$ such that $s_n$ is coprime to $n$?
[i]Russian Competition[/i]
Let $p\geq 3$ a prime number, $a$ and $b$ integers such that $\gcd(a, b)=1$. Let $n$ a natural number such that $p$ divides $a^{2^n}+b^{2^n}$, prove that $2^{n+1}$ divides $p-1$.
Let $m,n$ be positive integers and $p$ be a prime number.
Show that if $\frac{7^m + p \cdot 2^n}{7^m - p \cdot 2^n}$ is an integer, then it is a prime number.