Found problems: 988
Find all prime $p$, for each of which there are such natural $ x$ and $y$ such that $p^x = y^3 + 1$.
Find all pairs of integers $(x, y)$ satisfying the condition $12x^2 + 6xy + 3y^2 = 28(x + y)$.
Let $n\ge 1$ be a natural number. Determine all positive integer solutions of the equation
$$7 \cdot 4^n = a^2 + b^2 + c^2 + d^2.$$
Does there exist positive integers $a, b, c$ such that $4(ab - a - c^2) = b$?
Let $k,n,p$ be positive integers such that $p$ is a prime number, $k < 1000$ and $\sqrt{k} = n\sqrt{p}$.
a) Prove that if the equation $\sqrt{k + 100x} = (n + x)\sqrt{p}$ has a non-zero integer solution, then $p$ is a divisor of $10$.
b) Find the number of all non-negative solutions of the above equation.
Find all triples of positive integers $(m,p,q)$ such that $2^mp^2 + 27 = q^3$ and $p$ is a prime.
Find all natural numbers $m$ and $n$ such that $3^m+n!-1$ is the square of a natural number.
Determine all triples of positive integers $(a, b, n)$ that satisfy the following equation: $a! + b! = 2^n$
Prove that the equation $\prod_{cyc} (x_1-x_2)= \prod_{cyc} (x_1-x_3)$ has a solution in natural numbers where all $x_i$ are different.
Find all natural numbers $x$ and $y$ such that $x^y-y^x=1$ .
Find all triples $(a,b,p)$ of positive integers with $p$ prime and \[ a^p=b!+p. \]
[b]a)[/b] Find all solutions of the equation $p^2+q^2+r^2=pqr$, where $p,q,r$ are positive primes.\\
[b]b)[/b] Show that for every positive integer $N$, there exist three integers $a,b,c\geq N$ with $a^2+b^2+c^2=abc$.
For each positive integer $k$ find the number of solutions in nonnegative integers $x,y,z$ with $x\le y \le z$ of the equation
$$8^k=x^3+y^3+z^3-3xyz$$
Find all triples $(a, b, c)$ of positive integers for which $$\begin{cases} a + bc=2010 \\ b + ca = 250\end{cases}$$
Find all pairs $(x, y)$ of rational numbers such that $$xy^2=x^2+2x-3$$
Provided the equation $xyz = p^n(x + y + z)$ where $p \geq 3$ is a prime and $n \in \mathbb{N}$. Prove that the equation has at least $3n + 3$ different solutions $(x,y,z)$ with natural numbers $x,y,z$ and $x < y < z$. Prove the same for $p > 3$ being an odd integer.
The sequence $ (a_n)$ is defined by: $ a_0\equal{}a_1\equal{}1$ and $ a_{n\plus{}1}\equal{}14a_n\minus{}a_{n\minus{}1}$ for all $ n\ge 1$.
Prove that $ 2a_n\minus{}1$ is a perfect square for any $ n\ge 0$.
Suppose that $p$ and $q$ are two different positive integers and $x$ is a real number. Form the product $(x+p)(x+q).$ Find the sum $S(x,n) = \sum (x+p)(x+q),$ where $p$ and $q$ take values from 1 to $n.$ Does there exist integer values of $x$ for which $S(x,n) = 0.$
Find all pairs $x,y$ of natural numbers that satisfy the equation
$$x^2-xy+2x-3y=1997$$
Prove that there are infinite many positive integers $ n$ such that
$ n^2\plus{}1\mid n!$, and infinite many of those for which $ n^2\plus{}1 \nmid n!$.
Solve in integers the equation $x + y = x^2 - xy + y^2$.
Prove that for every positive integer $n$ there exist integers $a$ and $b,$ both greater than $1,$ such that $a ^ 2 + 1 = 2b ^ 2$ and $a - b$ is a multiple of $n.$
Let $p$ be a prime number in the form $p=4k+3$. Prove that if the numbers $x_0,y_0,z_0,t_0$ are solutions of the equation $x^{2p}+y^{2p}+z^{2p}=t^{2p}$, then at least one of them is divisible by $p$. [i](Plamen Koshlukov)[/i]
Solve in $\Bbb{N}^*$ the equation $$ 4^a \cdot 5^b - 3^c \cdot 11^d = 1.$$
Prove that there are no integers $x, y, z$ satisfying the equation $$x^2+y^2-z^2=xyz-2.$$
[i]Proposed by Navid Safaei[/i]