Found problems: 40
Find all positive integer pairs $(a,n)$ such that $\frac{(a+1)^n-a^n}{n}$ is an integer.
For each positive integer n, the number $R(n) = 11 ... 1$ is defined, which is made up of exactly $n$ digits equal to $1$. For example, $R(5) = 11111$. Let $n > 4$ be an integer for which, by writing all the positive divisors of $R(n)$, it is true that each written digit belongs to the set $\{0, 1\}$. Show that $n$ is a power of an odd prime number.
Clarification: A power of an odd prime number is a number of the form $p^a$, where $p$ is an odd prime number and $a$ is a positive integer.
Let $s(k)$ be the number of ways to express $k$ as the sum of distinct $2012^{th}$ powers, where order does not matter. Show that for every real number $c$ there exists an integer $n$ such that $s(n)>cn$.
[i]Alex Zhu.[/i]
Let $p>5$ be a prime number. Show that $p-4$ cannot be the fourth power of a prime number.
Given a positive integer $ n> 2 $. Prove that there exists a natural $ K $ such that for all integers $ k \ge K $ on the open interval $ ({{k} ^{n}}, \ {{(k + 1)} ^{n}}) $ there are $n$ different integers, the product of which is the $n$-th power of an integer.
Given that $x$, $y$ and $z$ are positive integers such that $x^3y^5z^6$ is a perfect 7th power of a positive integer, show that also $x^5y^6z^3$ is a perfect 7th power.
Determine whether there exists a positive integer $a$ such that $$2015a,2016a,\dots,2558a$$ are all perfect power.
Let $s(k)$ be the number of ways to express $k$ as the sum of distinct $2012^{th}$ powers, where order does not matter. Show that for every real number $c$ there exists an integer $n$ such that $s(n)>cn$.
[i]Alex Zhu.[/i]
Given the equation
$$
a^b\cdot b^c=c^a
$$
in positive integers $a$, $b$, and $c$.
[i](i)[/i] Prove that any prime divisor of $a$ divides $b$ as well.
[i](ii)[/i] Solve the equation under the assumption $b\ge a$.
[i](iii)[/i] Prove that the equation has infinitely many solutions.
[i](I. Voronovich)[/i]
Show that there is a set of $2002$ distinct positive integers such that the sum of one or more elements of the set is never a square, cube, or higher power.
Let $a,b,c,n$ be positive integers such that the following conditions hold
(i) numbers $a,b,c,a+b+c$ are pairwise coprime,
(ii) number $(a+b)(b+c)(c+a)(a+b+c)(ab+bc+ca)$ is a perfect $n$-th power.
Prove, that the product $abc$ can be expressed as a difference of two perfect $n$-th powers.
For all positive integers $a$ and $b$, we de ne $a @ b = \frac{a - b}{gcd(a, b)}$ .
Show that for every integer $n > 1$, the following holds:
$n$ is a prime power if and only if for all positive integers $m$ such that $m < n$, it holds that $gcd(n, n @m) = 1$.
Find all pairs $(m,n)$ of positive integers numbers with $m>1$ such that:
For any positive integer $b \le m$ that is not coprime with $m$, its posible choose positive integers $a_1, a_2, \cdots, a_n$ all coprimes with $m$ such that:
$$m+a_1b+a_2b^2+\cdots+a_nb^n$$
Is a perfect power.
Note: A perfect power is a positive integer represented by $a^k$, where $a$ and $k$ are positive integers with $k>1$
Find all positive integer pairs $(a,n)$ such that $\frac{(a+1)^n-a^n}{n}$ is an integer.
Let $p$ be a prime number and let $A$ be a set of positive integers that satisfies the following conditions:
(i) the set of prime divisors of the elements in $A$ consists of $p-1$ elements;
(ii) for any nonempty subset of $A$, the product of its elements is not a perfect $p$-th power.
What is the largest possible number of elements in $A$ ?
Consider integers $m\ge 2$ and $n\ge 1$.
Show that there is a polynomial $P(x)$ of degree equal to $n$ with integer coefficients such that $P(0),P(1),...,P(n)$ are all perfect powers of $m$ .
Examine whether exists $n \in N^*$, such that:
(a) $3n$ is perfect cube, $4n$ is perfect fourth power and $5n$ perfect fifth power
(b) $3n$ is perfect cube, $4n$ is perfect fourth power, $5n$ perfect fifth power and $6n$ perfect sixth power
For a positive integer $k,$ call an integer a $pure$ $k-th$ $power$ if it can be represented as $m^k$ for some integer $m.$ Show that for every positive integer $n,$ there exists $n$ distinct positive integers such that their sum is a pure $2009-$th power and their product is a pure $2010-$th power.
The Fibonacci numbers $(F_n)_{n=1}^{\infty}$ are defined as follows:
$$F_1 = F_2 = 1, F_n = F_{n-2} + F_{n-1}, n = 3, 4, ...$$
Assume $p$ is a prime greater than $3$. With $m$ being a natural number greater than $3$, find all $n$ numbers such that $F_n$ is divisible by $p^m$.
Find some natural number $a$ such that $2a$ is a perfect square, $3a$ is a perfect cube, $5a$ is the fifth power of some natural number.
Find an arithmetic progression of $2016$ natural numbers such that neither is a perfect power but its multiplication is a perfect power.
Clarification: A perfect power is a number of the form $n^k$ where $n$ and $k$ are both natural numbers greater than or equal to $2$.
Prove that for any natural $n>1$ there are infinitely many natural numbers $m$ such that for any nonnegative integers $k_1$,$k_2$, $...$,$k_m$, $$m \ne k_1^n+ k_2^n+... k_n^n,$$
An integer number $m\geq 1$ is [i]mexica[/i] if it's of the form $n^{d(n)}$, where $n$ is a positive integer and $d(n)$ is the number of positive integers which divide $n$. Find all mexica numbers less than $2019$.
Note. The divisors of $n$ include $1$ and $n$; for example, $d(12)=6$, since $1, 2, 3, 4, 6, 12$ are all the positive divisors of $12$.
[i]Proposed by Cuauhtémoc Gómez[/i]
For a positive integer $k,$ call an integer a $pure$ $k-th$ $power$ if it can be represented as $m^k$ for some integer $m.$ Show that for every positive integer $n,$ there exists $n$ distinct positive integers such that their sum is a pure $2009-$th power and their product is a pure $2010-$th power.
A pile of $2020$ stones is given. Arnaldo and Bernaldo play the following game: In each move, it is allowed to remove $1, 4, 16, 64, ...$ (any power of $4$) stones from the pile. They make their moves alternately, and the player who can no longer play loses. If Arnaldo is the first to play, who has the winning strategy?