Found problems: 916
Find all natural numbers $a$, $b$, $c$ and prime numbers $p$ and $q$, such that:
$\blacksquare$ $4\nmid c$
$\blacksquare$ $p\not\equiv 11\pmod{16}$
$\blacksquare$ $p^aq^b-1=(p+4)^c$
Find all triples of integers $(x, y, z)$ such that
$$x^2 + y^2 + z^2 = 16(x + y + z).$$
Find all positive integers $l$ for which the equation
\[
a^3+b^3+ab=(lab+1)(a+b)
\]
has a solution over positive integers $a,b$.
Determine all pairs of integers $(x, y)$ satisfying the equation
\[y(x + y) = x^3- 7x^2 + 11x - 3.\]
Show that there exist infinitely many positive integers $n$ such that $n^{2}+1$ divides $n!$.
Prove that there are no positive integers $x, y$ such that: $(x + 1)^2 + (x + 2)^2 +...+ (x + 9)^2 = y^2$
Find all triples of positive integers $(x, y, z)$ with $$\frac{xy}{z}+ \frac{yz}{x}+\frac{zx}{y}= 3$$
The natural numbers $p, q$ satisfy the relation $p^p + q^q = p^q + q^p$. Prove that $p = q$.
Does there exist an integer such that its cube is equal to $3n^2 + 3n + 7,$ where $n$ is an integer.
Let $A, B$, and $C$ be three points on the edge of a circular chord such that $B$ is due west of $C$ and $ABC$ is an equilateral triangle whose side is $86$ meters long. A boy swam from $A$ directly toward $B$. After covering a distance of $x$ meters, he turned and swam westward, reaching the shore after covering a distance of $y$ meters. If $x$ and $y$ are both positive integers, determine $y.$
Determine all pairs $(x, y)$ of positive integers such that for $d = gcd(x, y)$ the equation $$xyd = x + y + d^2$$
holds.
[i](Walther Janous)[/i]
Find all integer solutions $(x,y,z)$ of the equation $xy+yz+zx-xyz = 2$.
Find all integer solutions to the equation $y^k = x^2 + x$, where $k$ is a natural number greater than $1$.
Determine all positive integers $n$ for which the equation \[x^{n}+(2+x)^{n}+(2-x)^{n}= 0\] has an integer as a solution.
Find all pairs of integers $x,y$ for which
\[x^3+x^2+x=y^2+y.\]
Find all natural numbers $n$ such that the equation $x^2 + y^2 + z^2 = nxyz$ has solutions in positive integers
Solve in $ \mathbb{Z}^2 $ the equation: $ x^2\left( 1+x^2 \right) =-1+21^y. $
[i]Lucian Petrescu[/i]
Find all pairs $(m,n)$ of integers that satisfy the equation \[(m-n)^{2}=\frac{4mn}{m+n-1}.\]
Find all positive intger solutions of $3^x+29=2^y$.
Solve in integers the equation
\[ x^2+xy+y^2 = \left(\frac{x+y}{3}+1\right)^3. \]
Find all solutions $x,y,z$ in the positive integers of the equation $$3^x -5^y = z^2$$
Find all $(x, y, z, n) \in {\mathbb{N}}^4$ such that $ x^3 +y^3 +z^3 =nx^2 y^2 z^2$.
Determine all pairs $(p,m)$ consisting of a prime number $p$ and a positive integer $m$,
for which $p^3 + m(p + 2) = m^2 + p + 1$ holds.
Find all triples of positive integers $(a, b, c)$ such that $$(2^a-1)(3^b-1)=c!.$$
Prove that there exists infinitely many positive integers $n$ such that $n, n+1$, and $n+2$ can be written as the sum of two perfect squares.