Found problems: 988
Solve equation $x(x+1) = y(y+4)$ where $x$, $y$ are positive integers
Determine all integers $ a$ and $ b$, so that $ (a^3\plus{}b)(a\plus{}b^3)\equal{}(a\plus{}b)^4$
For every sequence $p_1<p_2<\cdots<p_8$ of eight prime numbers, determine the largest integer $N$ for which the following equation has no solution in positive integers $x_1,\ldots,x_8$:
$$p_1\, p_2\, \cdots\, p_8
\left( \frac{x_1}{p_1}+ \frac{x_2}{p_2}+ ~\cdots~ +\frac{x_8}{p_8} \right)
~~=~~ N $$
[i]Proposed by Gerhard Woeginger, Austria[/i]
Find all positive integers $n$ such that equation $$3a^2-b^2=2018^n$$ has a solution in integers $a$ and $b$.
Prove that if $a, b, c, d$ are integers such that $d=( a+\sqrt[3]{2}b+\sqrt[3]{4}c)^{2}$ then $d$ is a perfect square.
Let $p$ be a prime number. Find all pairs of coprime positive integers $(m,n)$ such that $$
\frac{p+m}{p+n}=\frac{m}{n}+\frac{1}{p^2}.$$
Solve in integer:$x^6+x^2y=y^3+2y^2$
Find all positive integers m, n such that $m^3 - n^3 = 999$.
Find all the integer solutions $(x,y,z)$ of the equation $(x + y + z)^5 = 80xyz(x^2 + y^2 + z^2)$,
Solve $y^5 - x^2 = 4$ in integers numbers $x,y$.
Find all triples of primes $(p, q, r)$ such that $p^q=2021+r^3$.
Three rational number $x,p,q$ satisfy $p^2-xq^2$=1. Prove that there are integers $a,b$ such that $p=\frac{a^2+xb^2}{a^2-xb^2}$ and $q=\frac{2ab}{a^2-xb^2}$.
Let $p, q, r$ be prime numbers and $t, n$ be natural numbers such that $p^2 +qt =(p + t)^n$ and $p^2 + qr = t^4$ .
a) Show that $n < 3$.
b) Determine all the numbers $p, q, r, t, n$ that satisfy the given conditions.
Find all non negative integers $(a, b)$ such that
$$a + 2b - b^2 =\sqrt{2a + a^2 + |2a + 1 - 2b|}.$$
Find the total number of integer solutions of the equation $$x^5-y^2=4$$
[i](Erken)[/i]
Let $a$, $b$, $c$ be given integers, $a>0$, $ac-b^2=p$ a squarefree positive integer. Let $M(n)$ denote the number of pairs of integers $(x, y)$ for which $ax^2 +bxy+cy^2=n$. Prove that $M(n)$ is finite and $M(n)=M(p^{k} \cdot n)$ for every integer $k \ge 0$.
How many pairs of positive integers $(x, y)$ are there, those satisfy the identity $2^x - y^2 = 4$?
(A): $1$ (B): $2$ (C): $3$ (D): $4$ (E): None of the above.
We say that a subset $A$ of $\mathbb N$ is good if for some positive integer $n$, the equation $x-y=n$ admits infinitely many solutions with $x,y\in A$. If $A_1,A_2,\ldots,A_{100}$ are sets whose union is $\mathbb N$, prove that at least one of the $A_i$s is good.
Determine if there is any triple of nonnegative integers, not necessarily different, $(a, b, c)$ such that:
$$a^3 + b^3 + c^3 = 2016$$
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.
Find the largest real number $a$ such that \[\left\{ \begin{array}{l}
x - 4y = 1 \\
ax + 3y = 1\\
\end{array} \right.
\] has an integer solution.
Determine all positive integers $a$, $b$ and $c$ which satisfy the equation
$$a^2+b^2+1=c!.$$
[i]Proposed by Nikola Velov[/i]
Find all the integer solutions of the equation $ m^4 + 2n^2 = 9mn$.
For all integral values of parameter $t$, find all integral solutions $(x,y)$ of the equation
$$ y^2 = x^4-22x^3+43x^2+858x+t^2+10452(t+39)$$ .
Find the natural numbers $x, y, z$ that verify the equation: $$2^x + 3 \cdot 11^y =7^z$$