Found problems: 364
Let $Q$ be a set of prime numbers, not necessarily finite. For a positive integer $n$ consider its prime factorization: define $p(n)$ to be the sum of all the exponents and $q(n)$ to be the sum of the exponents corresponding only to primes in $Q$. A positive integer $n$ is called [i]special[/i] if $p(n)+p(n+1)$ and $q(n)+q(n+1)$ are both even integers. Prove that there is a constant $c>0$ independent of the set $Q$ such that for any positive integer $N>100$, the number of special integers in $[1,N]$ is at least $cN$.
(For example, if $Q=\{3,7\}$, then $p(42)=3$, $q(42)=2$, $p(63)=3$, $q(63)=3$, $p(2022)=3$, $q(2022)=1$.)
A triple integer $(a, b, c)$ is called [i]brilliant [/i] when it satisfies:
(i) $a> b> c$ are prime numbers
(ii) $a = b + 2c$
(iii) $a + b + c$ is a perfect square number
Find the minimum value of $abc$ if triple $(a, b, c)$ is [i]brilliant[/i].
Let $p, q, r$ and $s$ be four prime numbers such that $$5 <p <q <r <s <p + 10.$$
Prove that the sum of the four prime numbers is divisible by $60$.
(Walther Janous)
Let $p_1,p_2,\ldots,p_n$ be distinct primes greater than $3$. Show that $2^{p_1p_2\cdots p_n}+1$ has at least $4^n$ divisors.
Find all the pairs of positive integers $(x,p)$ such that p is a prime, $x \leq 2p$ and $x^{p-1}$ is a divisor of $ (p-1)^{x}+1$.
Let the sequence ($a_n$) be defined by $a_n = n^6 +5n^4 -12n^2 -36, n \ge 2$.
(a) Prove that any prime number divides some term in this sequence.
(b) Prove that there is a positive integer not dividing any term in the sequence.
(c) Determine the least $n \ge 2$ for which $1989 | a_n$.
Find all triples $(p, x, y)$ consisting of a prime number $p$ and two positive integers $x$ and $y$ such that $x^{p -1} + y$ and $x + y^ {p -1}$ are both powers of $p$.
[i]Proposed by Belgium[/i]
Let $n \ge 5$ be an integer. Prove that $n$ is prime if and only if for any representation of $n$ as a sum of four positive integers $n = a + b + c + d$, it is true that $ab \ne cd$.
Find all pairs of natural numbers $(p, n)$ with $p$ prime such that $p^6+p^5+n^3+n=n^5+n^2$.
Find all the prime numbers $p, q$ and $r$ such that $p^2 + 1 = 74 (q^2 + r^2)$.
Let $p_k$ be the $k$-th prime number. Find the remainder when $\sum_{k=2}^{2550}p_k^{p_k^4-1}$ is divided by $2550$.
Let $n \ge 2$ be an integer and $p_1 < p_2 < ... < p_n$ prime numbers. Prove that there exists an integer $k$ relatively prime with $p_1p_2... p_n$ and such that $gcd (k + p_1p_2...p_i, p_1p_2...p_n) = 1$ for all $i = 1, 2,..., n - 1$.
Malik Talbi
Find the smallest prime $q$ such that $$q = a_1^2 + b_1^2 = a_2^2 + 2b_2^2 = a_3^2 + 3b_3^2 = ... = a_{10}^ 2 + 10b_{10}^2$$ where $a_i, b_i(i = 1, 2, ...,10)$ are positive integers
Let $a,b,c$ be positive integers such that the numbers $k=b^c+a, l=a^b+c, m=c^a+b$ to be prime numbers. Prove that at least two of the numbers $k,l,m$ are equal.
It is said that a positive integer $n > 1$ has the property ($p$) if in its prime factorization $n = p_1^{a_1} \cdot ... \cdot p_j^{a_j}$ at least one of the prime factors $p_1, ... , p_j$ has the exponent equal to $2$.
a) Find the largest number $k$ for which there exist $k$ consecutive positive integers that do not have the property ($p$).
b) Prove that there is an infinite number of positive integers $n$ such that $n, n + 1$ and $n + 2$ have the property ($p$).
Let us call the number of factors in the prime decomposition of an integer $n > 1$ the complexity of $n$. For example, [i]complexity [/i] of numbers $4$ and $6$ is equal to $2$. Find all $n$ such that all integers between $n$ and $2n$ have complexity
a) not greater than the complexity of $n$.
b) less than the complexity of $n$.
(Boris Frenkin)
On a blackboard a finite number of integers greater than one are written. Every minute, Nordi additionally writes on the blackboard the smallest positive integer greater than every other integer on the blackboard and not divisible by any of the numbers on the blackboard. Show that from some point onwards Nordi only writes primes on the blackboard.
Find all positive integers $n>2$ such that
$$ n! \mid \prod_{ p<q\le n, p,q \, \text{primes}} (p+q)$$
Find all $n$ for which there are $n$ consecutive integers whose sum of squares is a prime.
Prove that there are infinitely many primes representable in the form $m^2+mn+n^2$ for some integers $m,n$ .
Initially a number $6$ is written on a blackboard. At $n$-th step an integer $k$ on the blackboard is replaced by $k+gcd(k,n)$. Prove that at each step the number on the blackboard increases either by $1$ or by a prime number.
Show that if the difference of two positive cube numbers is a positive prime, then this prime number has remainder $1$ after division by $6$.
Let $n$ be an integer that is greater than $1$. Prove that the following two statements are equivalent:
(A) There are positive integers $a, b$ and $c$ that are not greater than $n$ and for which that polynomial $ax^2 + bx + c$ has two different real roots $x_1$ and $x_2$ with $| x_2- x_1 | \le \frac{1}{n}$
(B) The number $n$ has at least two different prime divisors.
Proof that for every primes $p$, $q$
\[p^{q^2-q+1}+q^{p^2-p+1}-p-q\]
is never a perfect square.
[i]Proposed by chengbilly[/i]
We say that a positive integer is [i]decomposed [/i] if it is prime and also If a line is drawn separating it into two numbers, those two numbers are never composite. For example 1997 is [i]decomposed [/i] since it is prime, it is divided into: $1$, $997$; $19$, $97$; $199$, $7$ and none of those numbers are compound. How many [i]decomposed [/i] numbers are there between $2000$ and $3000$?