Found problems: 988
Let $a$ be a positive integer. Prove that for any pair $(x,y)$ of integer solutions of equation $$x(y^2-2x^2)+x+y+a=0$$ we have: $$|x| \leqslant a+\sqrt{2a^2+2}$$
If $a, b, c \ge 4$ are integers, not all equal, and $4abc = (a+3)(b+3)(c+3)$ then what is the value of $a+b+c$ ?
Find all integer solutions to the equation $7x^2y^2 + 4x^2 = 77y^2 + 1260$.
Show that there are no positive integers $a$ und $b$ such that $4a(a + 1) = b(b + 3)$
Determine all odd primes $p$ and $q$ such that the equation $x^p + y^q = pq$ at least one solution $(x, y)$ where $x$ and $y$ are positive integers.
Show that the equation $14x^2 +15y^2 = 7^{2000}$ has no integer solutions.
Find the sum of the numbers written with two digits $\overline{ab}$ for which the equation $3^{x + y} =3^x + 3^y + \overline{ab}$ has at least one solution $(x, y)$ in natural numbers.
How many integer pairs $(x,y)$ satisfy $x^2+4y^2-2xy-2x-4y-8=0$?
Determine all pairs $(x, y)$ of natural numbers satisfying the equation $5^x=y^4+4y+1$.
A positive integer $a$ is called a [i]double number[/i] if it has an even number of digits (in base 10) and its base 10 representation has the form $a = a_1a_2 \cdots a_k a_1 a_2 \cdots a_k$ with $0 \le a_i \le 9$ for $1 \le i \le k$, and $a_1 \ne 0$. For example, $283283$ is a double number. Determine whether or not there are infinitely many double numbers $a$ such that $a + 1$ is a square and $a + 1$ is not a power of $10$.
Find all pairs of positive integers $x,y$ such that $\frac{4}{x}+\frac{2}{y}=1$
Solve in nonnegative integers the equation $5^t + 3^x4^y = z^2$.
Determine all pairs $(a, b)$ of integers which satisfy the equality $\frac{a + 2}{b + 1} +\frac{a + 1}{b + 2} = 1 +\frac{6}{a + b + 1}$
Determine all pairs $(x,y)$ of integers satisfying
\[(x+2)^4-x^4=y^3.\]
Find all pairs of primes $(p, q)$ for which $p-q$ and $pq-q$ are both perfect squares.
Prove that for every $n$ there exists a solution of the equation
$$a^2 +b^2 +c^2 = 3abc$$
in natural numbers $a,b,c$ greater than $n$.
Find the positive integers $n$ that are not divisible by $3$ if the number $2^{n^2-10}+2133$ is a perfect cube.
[hide="Note"]
[color=#BF0000]The wording of this problem is perhaps not the best English. As far as I am aware, just solve the diophantine equation $x^3=2^{n^2-10}+2133$ where $x,n \in \mathbb{N}$ and $3\nmid n$.[/color][/hide]
Find all triplets $(a,b,p)$ where $a,b$ are positive integers and $p$ is a prime number satisfying: $\frac{1}{p}=\frac{1}{a^2}+\frac{1}{b^2}$
Suppose that $a, b$, and $p$ are integers such that $b \equiv 1 \; \pmod{4}$, $p \equiv 3 \; \pmod{4}$, $p$ is prime, and if $q$ is any prime divisor of $a$ such that $q \equiv 3 \; \pmod{4}$, then $q^{p}\vert a^{2}$ and $p$ does not divide $q-1$ (if $q=p$, then also $q \vert b$). Show that the equation \[x^{2}+4a^{2}= y^{p}-b^{p}\] has no solutions in integers.
Find all triplets $(x, y, p)$ of positive integers such that $p$ is a prime number and $\frac{xy^3}{x+y}=p.$
Find all positive integers $m$ and $n$ for which \[1!+2!+3!+\cdots+n!=m^{2}\]
Solve the following Diophantines.
[b]a)[/b] $ x^2+y^2=6z^2 $
[b]b)[/b] $ x^2+y^2-2x+4y-1=0 $
[i]Dan Negulescu[/i]
Solve the following system
\[
\left\{ \begin{array}{l}
xyzu-x^3=9 \\
x+yz=\dfrac{3}{2}u \\
\end{array} \right.
\]
in positive integers $x$, $y$, $z$ and $u$.
Find all solutions to
\[ (abcde)^2 = a^2+b^2+c^2+d^2+e^2+f^2. \]
in integers.
Proposed by [i]Seongjin Shim[/i]
Solve for prime numbers $p, q, r$ : $$\frac{p}{q} - \frac{4}{r + 1}= 1$$