Found problems: 436
Determine all triples $(x, y, z)$ of positive integers $x > y > z > 0$, such that $x^2 = y \cdot 2^z + 1$
(a) Find a pair of integers (x,y) such that $15x^2 +y^2 = 2^{2000}$
(b) Does there exist a pair of integers $(x,y)$ such that $15x^2 + y^2 = 2^{2000}$ and $x$ is odd?
Show that for any positive integer $n$ there exists an integer $m > 1$ such that $(\sqrt2-1)^n=\sqrt{m}-\sqrt{m-1}$.
Are there natural numbers $(m,n,k)$ that satisfy the equation $m^m+ n^n=k^k$ ?
Find all primes $p,q$ such that $p ^3-q^7=p-q$.
Determine all the ways in which the fraction $\frac{1}{11}$ can be written as $\frac{1}{n}+\frac{1}{m}$ , where $n$ and $m$ are two different positive integers.
Find all pairs $(m,n)$ of positive integers such that $m^2 + n^2 = 3(m + n)$.
Find all primes $p, q$ such that $p + q = (p-q)^3$.
Let $x,y,z$ be integer numbers satisfying the equality $yx^2+(y^2-z^2)x+y(y-z)^2=0$
a) Prove that number $xy$ is a perfect square.
b) Prove that there are infinitely many triples $(x,y,z)$ satisfying the equality.
I.Voronovich
Let $N$ be the number of ordered pairs $(x,y)$ of integers such that $x^2+xy+y^2 \le 2007$.
Remember, integers may be positive, negative, or zero!
(a) Prove that $N$ is odd.
(b) Prove that $N$ is not divisible by $3$.
Find all integer $n$ such that the equation $2x^2 + 5xy + 2y^2 = n$ has integer solution for $x$ and $y$.
Find all pairs $(p, q)$ of prime numbers such that $$p(p^2 -p - 1) = q(2q + 3).$$
Integers $a, b, c$ satisfy $a+b-c=1$ and $a^2+b^2-c^2=-1$. What is the sum of all possible values of $a^2+b^2+c^2$ ?
Solve the system of equations in integers
$$x + y + z = 3$$
$$x^3 + y^3 + z^3 = 3$$
For which positive integers $m$ does the equation: $$(ab)^{2015} = (a^2 + b^2)^m$$ have positive integer solutions?
How many pairs $(a, b)$ of positive integers $a,b$ solutions of the equation $(4a-b)(4b-a )=1770^n$ exist , if $n$ is a positive integer?
If $a$ and $b$ are integers and if the solutions of the system of equations
$$y - 2x - a = 0$$
$$y^2 - xy + x^2 - b = 0$$
are rational, prove that the solutions are integers.
Determine whether exist a prime number $p$ and natural number $n$ such that $n^2 + n + p = 1996$.
Let $A$ be the set of all triples $(x, y, z)$ of positive integers satisfying $2x^2 + 3y^3 = 4z^4$ .
a) Show that if $(x, y, z) \in A$ then $6$ divides all of $x, y, z$.
b) Show that $A$ is an infinite set.
Find all integers $x,y,z$ such that $x^3 +5y^3 = 9z^3$.
Find all pairs of integers $(a, b)$ such that $a^2 + ab + b^2 = 1$
Find all positive integers $(m, n)$ such that $3 \cdot 2^n + 1 = m^2$.
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?
Prove that there is no pair of relatively prime positive integers $(a, b)$ that satisfy the equation
$$a^3 + 2017a = b^3 -2017b.$$
Determine, with proof, all the integer solutions of the equation $x^3 + 2y^3 + 4z^3 - 6xyz = 1$.