Found problems: 521
Prove that the product of all integers from $2^{1917} +1$ up to $2^{1991} -1$ is not the square of an integer.
(V. Senderov, Moscow)
Does there exist a natural number $n$, for which $n.2^{2^{2014}}-81-n$ is a perfect square?
Let $f = aX^2 + bX+ c \in Z[X]$ be a polynomial such that for every positive integer $n$,$ f(n )$ is a perfect square. Prove that $f = g^2$ for some polynomial $g \in Z[X]$.
It is given that n is a positive integer such that both numbers $2n + 1$ and $3n + 1$ are complete squares. Is it true that $n$ must be divisible by $40$ ? Justify your answer.
The positive integers $a$, $b$, $c$ are such that $\frac{a+b}{b+c}$ is the square of a rational number, and $ab+bc+ca$ is a prime number. Find all possible values of $\frac{a+b}{b+c}$.
Let $N = 11\cdots 122 \cdots 25$, where there are $n$ $1$s and $n+1$ $2$s. Show that $N$ is a perfect square.
Prove that if $n$ is a positive integer then $2 (n^2 + 1) - n$ is not a square of an integer.
Show that, for every positive integer $x$, there is a positive integer $y\in \{2, 5, 13\}$ such that $xy - 1$ is not a perfect square.
Is it possible for the least common multiple of five consecutive positive integers to be a perfect square?
For each positive integer $\,n,\;S(n)\,$ is defined to be the greatest integer such that, for every positive integer $\,k\leq S(n),\;n^{2}\,$ can be written as the sum of $\,k\,$ positive squares.
[b]a.)[/b] Prove that $\,S(n)\leq n^{2}-14\,$ for each $\,n\geq 4$.
[b]b.)[/b] Find an integer $\,n\,$ such that $\,S(n)=n^{2}-14$.
[b]c.)[/b] Prove that there are infintely many integers $\,n\,$ such that $S(n)=n^{2}-14.$
Determine the smallest number $n \in Z_+$, which can be written as $n = \Sigma_{a\in A}a^2$, where $A$ is a finite set of positive integers and $\Sigma_{a\in A}a= 2014$.
In other words: what is the smallest positive number which can be written as a sum of squares of different positive integers summing to $2014$?
Find all odd integers $n$ for which $\frac{2^{\phi (n)}-1}{n}$ is a perfect square.
For a positive integer $a$, define the sequence ($x_n$) by $x_1 = x_2 = 1$ and $x_{n+2 }= (a^4 +4a^2 +2)x_{n+1} -x_n -2a^2$ , for n $\ge 1$. Show that $x_n$ is a perfect square and that for $n > 2$ its square root equals the first entry in the matrix $\begin{pmatrix}
a^2+1 & a \\
a & 1
\end{pmatrix}^{n-2}$
We will say that a positive integer is a [i]winner [/i] if it can be written as the sum of a perfect square plus a perfect cube. For example, $33$ is a winner because $33=5^2+2^3$ .
Gabriel chooses two positive integers, r and s, and Germán must find $2005$ positive integers $n$ such that for each $n$, the numbers $r+n$ and $s+n$ are winners.
Prove that Germán can always achieve his goal.
Prove that no number of the form $a^3+3a^2+a$, for a positive integer $a$, is a perfect square.
For a positive integer $n$, let $s(n)$ and $c(n)$ be the number of divisors of $n$ that are perfect squares and perfect cubes respectively. A positive integer $n$ is called fair if $s(n)=c(n)>1$. Find the number of fair integers less than $100$.
Find all natural numbers $n$ for which $2^8 +2^{11} +2^n$ is a perfect square.
Show that, if $m$ and $n$ are non-zero integers of like parity, and $n^2 -1$ is divisible by $m^2 - n^2 + 1$, then $m^2 - n^2 + 1$ is the square of an integer.
Amer. Math. Monthly
Let \( x \), \( y \), \( p \) be positive integers that satisfy the equation \( x^4 = p + 9y^4 \), where \( p \) is a prime number. Show that \( \frac{p^2 - 1}{3} \) is a perfect square and a multiple of 16.
A positive integer $m$ is called a [i]beautiful [/i] integer if that there exists a positive integer $n$ such that $m = n^2+ n + 1$. Prove that there are infinitely many [i]beautiful [/i] integers with square factors, and the square factors of different beautiful integers are relative prime.
Let $a$ and $b$ be two positive integers. Prove that the integer
\[a^2+\left\lceil\frac{4a^2}b\right\rceil\]
is not a square. (Here $\lceil z\rceil$ denotes the least integer greater than or equal to $z$.)
[i]Russia[/i]
The product $1\times 2\times 3\times ...\times n$ is written on the board. For what integers $n \ge 2$, we can add exclamation marks to some factors to convert them into factorials, in such a way that the final product can be a perfect square?
Calculate the product of all positive integers less than $100$ and having exactly three positive divisors. Show that this product is a square.
$ M$ is a subset of $ \{1, 2, 3, \ldots, 15\}$ such that the product of any three distinct elements of $ M$ is not a square. Determine the maximum number of elements in $ M.$
Find all pairs $(m, n)$ of natural numbers for which the numbers $m^2 - 4n$ and $n^2 - 4m$ are both perfect squares.