Prove that there are at least $2005$ pairs of $(x, y)$ of non negative integers x and y that satisfy $x^2+y^2 = N,$ for some positive integer $N$.
1001
Number of integer solutions of $x^2 + y^2 = k$
116921
Find the unique pair of real numbers $(x,y)$ that satisfy $P(x)Q(y)=28$
35066
Nature and number of solutions to $xy=x+y$
71361
Prove that $x^x+y^y=z^z$ doesn't have integer solutions
4061
Proving there are infinitely many pairs of square-full consecutive integers
142513
Prove that $2xy\mid x^2+y^2-x$ implies $x$ is a perfect square.
54369
Proof of $n^2 \leq 2^n$.
33314
How many non-negative integer solutions does $x_1+x_2+\cdots+x_n=A$ have?
114404
For which positive integer values of $n$ does $a|mn-b?$
105347
How many nonnegative integer solutions are there to the pair of equations $x_1+x_2+…+x_6=20$ and $x_1+x_2+x_3=7$?
149071
Let $d$ be any positive integer not equal to $2, 5,$ or $ 13$ , then $\exists a, b \in \{2, 5, 13, d\}$ such that $ab − 1$ is not a perfect square?
126820
How many positive integer solutions are there to the equality $x_1+x_2+...+x_r= n$?
145265
A non-zero function satisfying $g(x+y) = g(x)g(y)$ must be positive everywhere
44019
Show that for any prime $ p $, there are integers $ x $ and $ y $ such that $ p|(x^{2} + y^{2} + 1) $.
96185
Solutions of $x^2+y^2+z^2 = n$ in $\mathbb{Z}_2$.
56574
For any integer $a,b$ let $N_{a,b}$ denote the number of positive integer $x<1000$ satisfying $x= a( mod\;27)$ and $x=b(mod\;37)$. Then
5595
$x^y = y^x$ for integers $x$ and $y$