How many among the first $1000$ Fibonacci numbers are multiples of $11$?
For how many positive integers $m$ is there at least 1 positive integer $n$ such that $mn \le m + n$?
Using the digits 1, 2, 3, 4, 5, 6, 7, and 9, form 4 two-digit prime numbers, using each digit only once. What is the sum of the 4 prime numbers?
A set of tiles numbered 1 through 100 is modified repeatedly by the following operation: remove all tiles numbered with a perfect square, and renumber the remaining tiles consecutively starting with 1. How many times must the operation be performed to reduce the number of tiles in the set to one?
For each positive integer $n$, let $s(n)$ denote the number of ordered positive integer pair $(x, y)$ for which $\frac{1}{x} + \frac{1}{y} = \frac{1}{n}$ holds. Find all positive integers $n$ for which $s(n) = 5$.
Show that for any right triangle whose sides' lengths are all integers,
- one side's length must be a multiple of 3, and
- one side's length must be a multiple of 4, and
- one side's length must be a multiple of 5
Please note these sides may not be distinct. For example, in a 5-12-13 triangle, 12 is a multiple of both 3 and 4.
Let $a$, $b$, and $c$ form a geometric sequence. Can the last two digits of $N=a^3+b^3+c^3-3abc$ be 20?
Solve in integers the equation $x^2 + y^2 - 1 = 4xy$
Solve in $\textit{rational}$ numbers the equation $x^2 - dy^2 = 1$ where $d$ is an integer.
Let $x$ be a positive real number, and $\lfloor{x}\rfloor$ be the largest integer that not exceeding $x$. Prove that there exist infinity number of positive integers, $n$, such that $\lfloor{\sqrt{2}}\ n\rfloor$ is a perfect square.
Show that there are infinitely many integers $n$ such that $2n + 1$ and $3n + 1$ are perfect squares, and that such $n$ must be multiples of $40$.
Show that the equation $x^2 + y^3 = z^4$ has infinitely many integer solutions.
Find all $n\in\mathbb{N}$ such that $$\binom{n}{k-1} = 2 \binom{n}{k} + \binom{n}{k+1}$$
for some natural number $k < n$.
Prove that if $m=2+2\sqrt{28n^2 +1}$ is an integer for some $n\in\mathbb{N}$, then $m$ is a perfect square.
Prove that if the difference of two consecutive cubes is $n^2$, $n\in\mathbb{N}$, then $(2n-1)$ is a square.
If $n$ is an integer such that the values of $(3n+1)$ and $(4n+1)$ are both squares, prove that $n$ is a multiple of $56$.
Let $p$ be a prime. Prove that the equation $x^2-py^2 = -1$ has integral solution if and only if $p=2$ or $p\equiv 1\pmod{4}$.
Let integers $a$, $b$ and $c$ be the lengths of a right triangle's three sides, where $c > b > a$. Show that $\frac{(c-a)(c-b)}{2}$ must be a square number.
If $p$ is a prime of the form $4k+3$, show that exactly one of the equations $x^2-py^2=\pm 2$ has an integral solution.
Show that $3^n-2$ is a square only for $n=1$ and $n=3$.
Show that if $\frac{x^2+1}{y^2}+4$ is a perfect square, then this square equals $9$.
Solve in non-negative integers the equation $$x^3 + 2y^3 = 4z^3$$
Solve in nonnegative integers the equation $$2^x -1 = xy$$
Solve in integers the equation $x^2+y^2+z^2-2xyz=0$
For any given positive integer $n > 2$, show that there exists a right triangle with all sides' lengths are integers and one side's length equals $n$.