Found problems: 521
Let $\left\{ {{a}_{n}} \right\}_{n \geq 1}$ and $\left\{ {{b}_{n}} \right\}_{n \geq 1}$ be two infinite arithmetic progressions, each of which the first term and the difference are mutually prime natural numbers. It is known that for any natural $n$, at least one of the numbers $\left( a_n^2+a_{n+1}^2 \right)\left( b_n^2+b_{n+1}^2 \right) $ or $\left( a_n^2+b_n^2 \right) \left( a_{n+1}^2+b_{n+1}^2 \right)$ is an perfect square. Prove that ${{a}_{n}}={{b}_{n}}$, for any natural $n$ .
During the factoring class, Esmeralda observed that $1$, $3$ and $5$ can be written as the difference of two perfect squares, as can be seen:
$1 = 1^2 - 0^2$
$3 = 2^2 - 1^2$
$5 = 3^2 - 2^2$
a) Show that all numbers written in the form $2 * m + 1$ can be written as a difference of two perfect squares.
b) Show how to calculate the value of the expression $E = 1 + 3 + 5 + ... + (2m + 1)$.
c) Esmeralda, happy with what she discovered, decided to look for other ways to write $2019$ as the difference of two perfect squares of positive integers. Determine how many ways it can do what you want.
Each integer has been colored in one of three colors. Prove that exist two different numbers of the same color, whose difference is a perfect square.
Show that the set of positive integers that cannot be represented as a sum of distinct perfect squares is finite.
Are there $10$ different integers such that all the sums made up of $9$ of them are perfect squares?
Let $a$, $b$, and $c $ be integers greater than zero. Show that the numbers $$2a ^ 2 + b ^ 2 + 3 \,\,, 2b ^ 2 + c ^ 2 + 3\,\,, 2c ^ 2 + a ^ 2 + 3 $$ cannot be all perfect squares.
Let $n$ be a positive integer.
a) Prove that the number $(2n + 1)^3 - (2n - 1)^3$ is the sum of three perfect squares.
b) Prove that the number $(2n+1)^3-2$ is the sum of $3n-1$ perfect squares greater than $1$.
Let $x,y,z$ be integer numbers satisfying the equality $yx^2+(y^2-z^2)x+y(y-z)^2=0$
a) Prove that number $xy$ is a perfect square.
b) Prove that there are infinitely many triples $(x,y,z)$ satisfying the equality.
I.Voronovich
Let $ a, b \in Z $, prove that if the expression $ a \cdot 2013^n + b $ is a perfect square for all natural $n$, then $ a $ is zero.
Prove that $2551 \cdot 543^n -2008\cdot 7^n$ is never a perfect square, where $n$ varies over the set of positive integers
Determine all positive integers $N$ such that
$$2^N-2N$$
is a perfect square.
(Walther Janous)
Prove that there are infinitely many sets of four positive integers so that the sum of the squares of any three elements is a perfect square.
All expressions of the form $$\pm \sqrt1 \pm \sqrt2 \pm ... \pm \sqrt{100}$$ (with every possible combination of signs) are multiplied together. Prove that the result is:
(a) an integer;
(b) the square of an integer.
(A Kanel)
Integers $a, b$ and $c$ satisfy the condition $ab + bc + ca = 1$. Is it true that the number $(1+a^2)(1+b^2)(1+c^2)$ is a perfect square? Why?
Find all positive integers $n$ for which $1 - 5^n + 5^{2n+1}$ is a perfect square.
Can a number of the form $44\dots 41$, with an odd number of decimal digits $4$ followed by a digit $1$, be a perfect square?
Prove that if the integers $ a $ and $ b $ satisfy the equation
$$ 2a^2 + a = 3b^2 + b,$$
then the numbers $ a - b $ and $ 2a + 2b + 1 $ are squares of integers.
Let $n \ge 0$ be an integer. A sequence $a_0,a_1,a_2,...$ of integers is de fined as follows:
we have $a_0 = n$ and for $k \ge 1, a_k$ is the smallest integer greater than $a_{k-1}$ for which $a_k +a_{k-1}$ is the square of an integer.
Prove that there are exactly $\lfloor \sqrt{2n}\rfloor$ positive integers that cannot be written in the form $a_k - a_{\ell}$ with $k > \ell\ge 0$.
Find all natural numbers whose decimal notation consists of different digits of the same parity and which are perfect squares.
A natural number is called [i]bright [/i] if it is the sum of a perfect square and a perfect cube.
Prove that if $r$ and $s$ are any two positive integers, then
(a) there exist infinitely many positive integers $n$ such that both $r+n$ and $s+n$ are [i]bright[/i],
(b) there exist infinitely many positive integers $m$ such that both rm and sm are [i]bright[/i].
Let $a$ and $b$ be natural numbers such that $2a-b$, $a-2b$ and $a+b$ are all distinct squares. What is the smallest possible value of $b$ ?
For a positive integer $n$, let $\varphi(n)$ and $d(n)$ denote the value of the Euler phi function at $n$ and the number of positive divisors of $n$, respectively. Prove that there are infinitely many positive integers $n$ such that $\varphi(n)$ and $d(n)$ are both perfect squares.
[i]Finland, Olli Järviniemi[/i]
Let $a$ and $b$ be two positive integers such that $2ab$ divides $a^2+b^2-a$. Prove that $a$ is perfect square
Do there exist $1000 000$ distinct positive integers such that the sum of any collection of these numbers is never an exact square?
Let $n \ge 0$ be an integer. A sequence $a_0,a_1,a_2,...$ of integers is de fined as follows:
we have $a_0 = n$ and for $k \ge 1, a_k$ is the smallest integer greater than $a_{k-1}$ for which $a_k +a_{k-1}$ is the square of an integer.
Prove that there are exactly $\lfloor \sqrt{2n}\rfloor$ positive integers that cannot be written in the form $a_k - a_{\ell}$ with $k > \ell\ge 0$.