Found problems: 988
Find all pairs of positive integers $(x,y) $ for which $x^3 + y^3 = 4(x^2y + xy^2 - 5) .$
Let $n$ be a natural number. Solve in whole numbers the equation \[x^{n}+y^{n}=(x-y)^{n+1}.\]
Find integer solutions of $x^3+y^3-2xy+x+y+2=0$
Find all solutions for positive integers $(x,y,k,m)$ such that
\[ 20x^k+24y^m = 2024\]
with $k, m > 1$
Let $r$ be a positive integer and let $a_r$ be the number of solutions to the equation $3^x-2^y=r$ ,such that $0\leq x,y\leq 5781$ are integers. What is the maximal value of $a_r$?
Determine all prime numbers $p, q$ and $r$ with $p + q^2 = r^4$.
[i](Karl Czakler)[/i]
Find all positive prime numbers $p,q,r,s$ so that $p^2+2019=26(q^2+r^2+s^2)$.
Find all triples $(a, b, c)$ of positive integers such that $a \leq b$ and \[a!+b!=c^4+2024\]
[i]Proposed by Otgonbayar Uuye.[/i]
Find all pairs of prime numbers $(p, q)$ for which $7pq^2 + p = q^3 + 43p^3 + 1$
Are there integers $k$ and $m$ for which
$$\frac{(k-3)(k-2)(k-1)k+1}{(k+1)(k+2)(k+3)(k+4)+1}=m(m+1)+(m+1)(m+2)+(m+2)m \,\,
?$$
Find all the prime numbers $p$ and $q$ such that $ p^2+q=37q^2+p $.
Clarification: $1$ is not a prime number.
Two positive integers have the sum $2002$. Can $2002$ divide their product?
Find all couples of positive integers $m$ and $n$ such that
$$n!+5=m^3$$
Let $ a$ and $ b$ be two positive integers such that $ a \cdot b \plus{} 1$ divides $ a^{2} \plus{} b^{2}$. Show that $ \frac {a^{2} \plus{} b^{2}}{a \cdot b \plus{} 1}$ is a perfect square.
a) Given that integers $a$ and $b$ satisfy the equality $$a^2 - (b^2 - 4b + 1) a - (b^4 - 2b^3) = 0 \,\,\, (*)$$, prove that $b^2 + a$ is a square of an integer.
b) Do there exist an infinitely many of pairs $(a,b)$ satisfying (*)?
Let $p$ be a fixed prime. Find all pairs $(x,y)$ of positive numbers satisfying $p(x-y) = xy$.
Solve the equation in nonnegative integers:\\
$2^x=5^y+3$
Find all pairs $(x, y)$ of positive rational numbers such that $x^{2}+3y^{2}=1$.
Is there a positive integer $m$ such that the equation \[\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{abc}= \frac{m}{a+b+c}\] has infinitely many solutions in positive integers $a, b, c \;$?
Find all positive integers $n$ for which the equation $$(x^2 + y^2)^n = (xy)^{2016}$$ has positive integer solutions.
Solve the equation $p^3-q^3=pq^3-1$ in primes $p,q$.
Find all the positive integers $a, b, c$ such that $$a! \cdot b! = a! + b! + c!$$
Solve the system of equations in integers:
$$\begin{cases} a^3=abc+2a+2c \\ b^3=abc-c \\ c^3=abc-a+b \end{cases}$$
Show that the equation $x^7 + y^7 = {1998}^z$ has no solution in positive integers.
Find all integer solutions $(p, q, r)$ of the equation $r + p ^ 4 = q ^ 4$ with the following conditions:
$\bullet$ $r$ is a positive integer with exactly $8$ positive divisors.
$\bullet$ $p$ and $q$ are prime numbers.