Found problems: 10
Prove that for each positive integer $ n$ there exist $ n$ consecutive positive integers none of which is an integral 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.
Decide, whether exists positive rational number $w$, which isn't integer, such that $w^w$ is a rational number.
Ten different odd primes are given. Is it possible that for any two of them, the difference of their sixteenth powers to be divisible by all the remaining ones ?
Prove that for every positive integer $n$ there exists a positive integer $k$, such that each of the numbers $k, k^2, \dots, k^n$ have at least one block of $2022$ in their decimal representation.
For example, the numbers 4[b]2022[/b]13 and 544[b]2022[/b]1[b]2022[/b] have at least one block of $2022$ in their decimal representation.
Prove that for each positive integer $ n$ there exist $ n$ consecutive positive integers none of which is an integral power of a prime number.
Prove that for each positive integer $ n$ there exist $ n$ consecutive positive integers none of which is an integral power of a prime number.
Let $a$ and $b$ be two distinct natural numbers. It is known that $a^2+b|b^2+a$ and that $b^2+a$ is a power of a prime number. Determine the possible values of $a$ and $b$.
Show that the number $1989$ as well as each of its powers $1989^n$ ($n \in N$), can be expressed as a sum of two positive squares in at least two ways.
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?