Found problems: 988
Prove that there are infinitely many positive integers $ n$ for which all the prime divisors of $ n^{2}\plus{}n\plus{}1$ are not more then $ \sqrt{n}$.
[hide] Stronger one.
Prove that there are infinitely many positive integers $ n$ for which all the prime divisors of $ n^{3}\minus{}1$ are not more then $ \sqrt{n}$.[/hide]
Determine all integers solutions of the equation $x + x^3 = 5y^2$.
Prove that there are infinitely many quadruples of integers $(a,b,c,d)$ such that
\begin{align*}
a^2 + b^2 + 3 &= 4ab\\
c^2 + d^2 + 3 &= 4cd\\
4c^3 - 3c &= a
\end{align*}
[i]Travis Hance.[/i]
Prove that for all $n \in N$, $x^2 + y^2 = z^n$ has solutions with $x,y,z \in N$.
Let $x, y, p, n, k$ be positive integers such that $$x^n + y^n = p^k.$$
Prove that if $n > 1$ is odd, and $p$ is an odd prime, then $n$ is a power of $p$.
Find all pairs $(x, y)$ of natural numbers such that $$x + y = a^n,
x^2 + y^2 = a^m$$ for some natural $a, n, m$.
Let $m, n$ be odd prime numbers.
Find all pairs of integers numbers $a, b$ for which the system of equations:
$x^m+y^m+z^m=a$,
$x^n+y^n+z^n=b$
has many solutions in integers $x, y, z$.
Find all triples of primes $(p,q,r)$ satisfying $3p^{4}-5q^{4}-4r^{2}=26$.
Consider the equation $x^3 + y^3 + z^3 = 2$.
a) Prove that it has infinitely many integer solutions $x,y,z$.
b) Determine all integer solutions $x, y, z$ with $|x|, |y|, |z| \leq 28$.
Find all positive integers $a, b, c$, such that $(8a-5b)^2 + (3b-2c)^2 + (3c-7a)^2 = 2$.
Let $n$ be a positive integer and $a_1, a_2, \dots , a_{2n}$ mutually distinct integers. Find all integers $x$ satisfying
\[(x - a_1) \cdot (x - a_2) \cdots (x - a_{2n}) = (-1)^n(n!)^2.\]
Find all pairs of primes $(p, q)$ such that $p^3 - q^5 = (p + q)^2$
.
Let $ E(x,y)=\frac{x}{y} +\frac{x+1}{y+1} +\frac{x+2}{y+2} . $
[b]a)[/b] Solve in $ \mathbb{N}^2 $ the equation $ E(x,y)=3. $
[b]b)[/b] Show that there are infinitely many natural numbers $ n $ such that the equation $ E(x,y)=n $ has at least one solution in $ \mathbb{N}^2. $
Find all triples of primes $(p,q,r)$ satisfying $3p^{4}-5q^{4}-4r^{2}=26$.
Find the smallest value of the expression $|253^m - 40^n|$ over all pairs of positive integers $(m, n)$.
[i]Proposed by Oleksii Masalitin[/i]
Find the number of ordered triples of positive integers $(a, b, c)$ such that
\[6a + 10b + 15c = 3000.\]
Determine all prime numbers $ p $ and natural numbers $ x, y $ for which $ p^x-y^3 = 1 $.
(a) Let $k$ be an natural number so that the equation $ab + (a + 1) (b + 1) = 2^k$ does not have a positive integer solution $(a, b)$. Show that $k + 1$ is a prime number.
(b) Show that there are natural numbers $k$ so that $k + 1$ is prime numbers and equation $ab + (a + 1) (b + 1) = 2^k$ has a positive integer solution $(a, b)$.
Find all natural numbers $k$ such that there exist natural numbers $a_1,a_2,...,a_{k+1}$ with $ a_1!+a_2!+... +a_{k+1}!=k!$
Note that we do not consider $0$ to be a natural number.
Prove that for any integers $x$ and $y$ we have $x^5 + 3x^4y - 5x^3y^2 - 15x^2y^3 + 4xy^4 + 12y^5 \ne 33$.
Find all triples of positive integers $(x, y, z)$ such that \[(x+y)(1+xy)= 2^{z}.\]
Determine the pairs of positive integer numbers $m$ and $n$ that satisfy the equation $m^2=n^2 +m+n+2018$.
Find all pairs $(n, p)$ of positive integers such that $p$ is prime and
\[ 1 + 2 + \cdots + n = 3 \cdot (1^2 + 2^2 + \cdot + p^2). \]
Do there exist two right-angled triangles with integer length sides that have the lengths of exactly two sides in common?
Find a pair of relatively prime four digit natural numbers $A$ and $B$ such that for all natural numbers $m$ and $n$, $\vert A^m -B^n \vert \ge 400$.