Found problems: 60
For positive integers $n,a,b$, if $n=a^2 +b^2$, and $a$ and $b$ are coprime, then the number pair $(a,b)$ is called a [i]square split[/i] of $n$ (the order of $a, b$ does not count). Prove that for any positive $k$, there are only two square splits of the integer $13^k$.
Do there exist $10$ consecutive positive integers such that the sum of their squares is equal to the sum of squares of the next $9$ integers?
(Inspired by a diagram in an old text book)
Show that an integer $= 7 \mod 8$ cannot be sum of three squares.
Find the distinct positive integers $a, b, c,d$, such that the following conditions hold:
(1) exactly three of the four numbers are prime numbers;
(2) $a^2 + b^2 + c^2 + d^2 = 2018.$
Let $n$ be an integer. Show that a natural number $k$ can be found for which, the following applies with a suitable choice of signs: $$n = \pm 1^2 \pm 2^2 \pm 3^2 \pm ... \pm k^2$$
Find all pairs $(x, y)$ of integers satisfying the following condition:
each of the numbers $x^3 + y$ and $x + y^3$ is divisible by $x^2 + y^2$ .
(S Zlobin)
Is there any numbber $n$, such that the sum of its digits in the decimal notation is $1000$, and the sum of its square digits in the decimal notation is $1000000$?
Let $X$ be a set which consists from $8$ consecutive positive integers. Set $X$ is divided on two disjoint subsets $A$ and $B$ with equal number of elements. If sum of squares of elements from set $A$ is equal to sum of squares of elements from set $B$, prove that sum of elements of set $A$ is equal to sum of elements of set $B$.
Prove that the remainder of dividing the sum of two squares of integers by $4$ is different from $3$.
Show that a number $n(n+1)$ where $n$ is positive integer is the sum of 2 numbers $k(k+1)$ and $m(m+1)$ where $m$ and $k$ are positive integers if and only if the number $2n^2+2n+1$ is composite.
Find all $n$ for which there are $n$ consecutive integers whose sum of squares is a prime.
Positive integers $a,b,z$ satisfy the equation $ab=z^2+1$. Prove that there exist positive integers $x,y$ such that
$$\frac{a}{b}=\frac{x^2+1}{y^2+1}$$
Let $d$ be an even positive integer.
John writes the numbers $1^2 ,3^2 ,\ldots,(2n-1)^2 $ on the blackboard and then chooses three of them, let them be ${a_1}, {a_2}, {a_3}$, erases them and writes the number $1+ \displaystyle\sum_{1\le i<j\leq 3} |{a_i} -{a_j}|$
He continues until two numbers remain written on on the blackboard.
Prove that the sum of squares of those two numbers is different than the numbers $1^2 ,3^2 ,\ldots,(2n-1)^2$.
(Albania)
Show that the set of positive integers that cannot be represented as a sum of distinct perfect squares is finite.
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.
Determine all integers $n \geq 2$, satisfying
$$n=a^2+b^2,$$
where $a$ is the smallest divisor of $n$ different from $1$ and $b$ is an arbitrary divisor of $n$.
[i]Proposed by Walther Janous[/i]
Ten sportsmen have taken part in a table-tennis tournament (each pair has met once only, no draws). Let $xi$ be the number of $i$-th player victories, $yi$ -- losses. Prove that $$x_1^2 + ... + x_{10}^2 = y_1^2 + ... + y_{10}^2$$
We call a positive integer [i]good[/i ] if it doesn’t have a zero digit and the sum of the squares of its digits is a perfect square. For example, $122$ and $34$ are good and $304$ and $12$ are not not good. Prove that there exists a $n$-digit good number for every positive integer $n$.
For an integer $n$ let $M (n) = \{n, n + 1, n + 2, n + 3, n + 4\}$. Furthermore, be $S (n)$ sum of squares and $P (n)$ the product of the squares of the elements of $M (n)$. For which integers $n$ is $S (n)$ a divisor of $P (n)$ ?
Determine if there are $2018$ different positive integers such that the sum of their squares is a perfect cube and the sum of their cubes is a perfect square.
For each prime number $p$, determine the number of residue classes modulo $p$ which can
be represented as $a^2+b^2$ modulo $p$, where $a$ and $b$ are arbitrary integers.
[i](Daniel Holmes)[/i]
Find the smallest odd natural number $N$ such that $N^2$ is the sum of an odd number (greater than $1$) of squares of adjacent positive integers.
A positive integer $n$ is said to possess Property ($A$) if there exists a positive integer $N$ such that $N^2$ can be written as the sum of the squares of $n$ consecutive positive integers. Is it true that there are infinitely many positive integers which possess Property ($A$)? Justify your answer.
(As an example, the number $n = 2$ possesses Property ($A$) since $5^2 = 3^2 + 4^2$).
Let $p \geq 7$ be a prime number and $$S = \bigg\{jp+1 : 1 \leq j \leq \frac{p-5}{2}\bigg\}.$$ Prove that at least one element of $S$ can be written as $x^2+y^2$, where $x, y$ are integers.
Show that the set of positive integers that cannot be represented as a sum of distinct perfect squares is finite.