Found problems: 436
Determine all positive integers $n \ge 2$ which have a positive divisor $m | n$ satisfying $$n = d^3 + m^3.$$
where $d$ is the smallest divisor of $n$ which is greater than $1$.
Find:
a) Any quadruple of positive integers $(a, k, l, m)$ such that $a^k = a^l + a^m,$
b) Any quintuple of positive integers $(a, k, l, m, n)$ for which $a^k = a^l + a^m+a^n$
Show that there are only finitely many triples $(a, b, c)$ of positive integers such that $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{1}{1000}$.
Let $m,n > 1$ be integer numbers. Solve in positive integers $x^n+y^n = 2^m$.
Find the number of positive integer solutions to the equation $(x_1+x_2+x_3)^2(y_1+y_2) = 2548$.
Find all solutions of the equation $(a^a)^5 = b^b$ in positive integers.
Find all pairs $(x,y)$ of positive integers such that $x^{x+y} =y^{y-x}$.
Find a countable family of natural solutions to $ \frac{1}{a} +\frac{1}{b} +\frac{1}{ab}=\frac{1}{c} . $
Find all triples of integers $(a, b, c)$ satisfying $a^2 + b^2 + c^2 =3(ab + bc + ca).$
Find all pairs of integers $(a, b)$ such that $$a^2 + ab - b = 2018.$$
Let $p$ be a prime number and let $m$ and $n$ be positive integers with $p^2 + m^2 = n^2$.
Prove that $m> p$.
(Karl Czakler)
Find all pairs of integers $(x,y)$ satisfying the following equality $8x^2y^2 + x^2 + y^2 = 10xy$
Let $ p>3$ be a prime. If $ p$ divides $ x$, prove that the equation $ x^2-1=y^p$ does not have positive integer solutions.
Find all pairs of integers ($a, b$) such that $a^2 + b = b^{1999}$ .
Find all positive integers $(m, n)$ that satisfy $$m^2 =\sqrt{n} +\sqrt{2n + 1}.$$
Consider the set of solutions of the equation $$x^2+y^3=z^2.$$ in positive integers. Is it finite or infinite?
(Folklore)
How many pairs of positive integers $(x, y)$ are there, those satisfy the identity $2^x - y^2 = 1$?
(A): $1$ (B): $2$ (C): $3$ (D): $4$ (E): None of the above.
Find all triples $(a, b, c)$ of natural numbers $a, b$ and $c$, for which $a^{b + 20} (c-1) = c^{b + 21} - 1$ is satisfied.
(Walther Janous)
Solve the equation $xy =1997(x + y)$ in integers.
Find all nonnegative integers $a$ and $b$ that satisfy the equation $$3 \cdot 2^a + 1 = b^2.$$
Find all triples $(a, b, c)$ of integers that satisfy the equations
$ a + b = c$ and $a^2 + b^3 = c^2$
Find all integers $a$ and $b$ satisfying the equality $3^a - 5^b = 2$.
(I. Gorodnin)
Determine all positive integers $k$ for which there exist positive integers $n$ and $m, m\ge 2$, such that $3^k + 5^k = n^m$
Prove that there exist no positive integers $m$ and $n$ such that $m > 5$ and $(m - 1)! + 1 = m^n$.
Four positive integers $x,y,z$ and $t$ satisfy the relations
\[ xy - zt = x + y = z + t. \]
Is it possible that both $xy$ and $zt$ are perfect squares?