Found problems: 113
For all positive integer $n$, we consider the number $$a_n =4^{6^n}+1943$$ Prove that $a_n$ is dividible by $2013$ for all $n\ge 1$, and find all values of $n$ for which $a_n - 207$ is the cube of a positive integer.
Given a sequence of numbers $ 13, 25, 43, \ldots $ whose $ n $-th term is defined by the formula
$$a_n =3(n^2 + n) + 7$$
Prove that this sequence has the following properties:
1) Of every five consecutive terms of the sequence, exactly one is divisible by $ 5 $,
2( No term of the sequence is the cube of an integer.
Do there exist positive integers $a \ne b$ such that $ a+b$ is a perfect square and $a^3 +b^3$ is a fourth power of an integer?
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 $gcd(n, n @m) = 1$.
Prove that none of the numbers $11, 111, 1111, ...$ is a square number, cube number or higher power of a natural number.
Is it possible to pair up the numbers $0, 1, 2, 3,... , 61$ in such a way that when we sum each pair, the product of the $31$ numbers we get is a perfect f ifth power?
Does there exist an integer such that its cube is equal to $3n^2 + 3n + 7,$ where $n$ is an integer.
Find all natural numbers $n$ for which $n + 195$ and $n - 274$ are perfect cubes.
An infinite arithmetic progression whose terms are positive integers contains the square of an integer and the cube of an integer. Show that it contains the sixth power of an integer.
Let $x$ be the smallest positive integer for which $2x$ is the square of an integer, $3x$ is the third power of an integer, and $5x$ is the fifth power of an integer. Find the prime factorization of $x$.
(St. Wagner, Stellenbosch University)
Find all pairs $(x, y)$ of positive integers such that $(x + y)(x^2 + 9y)$ is the cube of a prime number.
Let $\mathbb{N}$ be the set of positive integers. Find all functions $f : \mathbb{N} \to \mathbb{N}$ such that for all $m, n \in \mathbb{N}$, \[m^2f(m) + n^2f(n) + 3mn(m + n)\] is a perfect cube.
Given that $x$ and $y$ are positive integers such that $xy^{433}$ is a perfect 2016-power of a positive integer, prove that $x^{433}y$ is also a perfect 2016-power.