Solve in positive integers the equation $3^x + 4^y = 5^z$ .
Solve in positive integers the equation $8^x + 15^y = 17^z$.
Find the number of ordered pairs of positive integer solutions $(m, n)$ to the equation $20m + 16n = 2016$
Let $P(x)$ be a polynomial with integer coefficients satisfying that both $P(0)$ and $P(1)$ are odd. Show that $P(x)$ has no integer zeros.
Find all prime numbers $p$ that can be written $p = x^4 + 4y^4$, where $x, y$ are positive integers.
Is $4^{545} + 545^{4}$ a prime?
Prove that if $n>1$, then $(n^4 + 4^n)$ is a composite number.
Compute $$\frac{(10^4+324)(22^4+324)(34^4+324)(46^4+324)(58^4+324)}{(4^4+324)(16^4+324)(28^4+324)(40^4+324)(52^4+324)}$$
Find the largest prime divisor of $25^2+72^2$
Calculate the value of $$\dfrac{2014^4+4 \times 2013^4}{2013^2+4027^2}-\dfrac{2012^4+4 \times 2013^4}{2013^2+4025^2}$$
For $k > 0$, let $I_k = 10\ldots 064$, where there are $k$ zeros between the $1$ and the $6$. Let $N(k)$ be the number of factors of $2$ in the prime factorization of $I_k$. What is the maximum value of $N(k)$?
Let $a, b$, and $c$ be three positive integers such that $\frac{1}{a^2}+\frac{1}{b^2}=\frac{1}{c^2}$. Find the sum of all possible $a$ where $a \le 100$.
A rectangular box has integer side lengths in the ratio $1: 3: 4$. Which of the following could be the volume of the box?
For some positive integer $n$, the number $110n^3$ has $110$ positive integer divisors, including $1$ and the number $110n^3$. How many positive integer divisors does the number $81n^4$ have?
How many ordered triples $(x,y,z)$ of positive integers satisfy $\text{lcm}(x,y) = 72, \text{lcm}(x,z) = 600$ and $\text{lcm}(y,z)=900$?
Find a square number which has two thousand and eighteen $6$s and some numbers of $0$s?
What is the tens digit of $2015^{2016}-2017?$
In how many ways can $345$ be written as the sum of an increasing sequence of two or more consecutive positive integers?
There are exactly $77,000$ ordered quadruplets $(a, b, c, d)$ such that $\gcd(a, b, c, d) = 77$ and $\operatorname{lcm}(a, b, c, d) = n$. What is the smallest possible value for $n$?
There exist some integers, $a$, such that the equation $(a+1)x^2 -(a^2+1)x+2a^2-6=0$ is solvable in integers. Find the sum of all such $a$.
How many ordered integeral triples $(x, y, z)$ have the property that each number is the product of the other two?
Let $n$ be a positive integer, and $d$ is a positive divisor of $2n^2$. Show that $(n^2+d)$ cannot be a square number.
Find the least positive integer $m$ such that $m^2 - m + 11$ is a product of at least four not necessarily distinct primes.
The sum of the three different positive unit fractions is $\frac{6}{7}$. What is the least number that could be the sum of the denominators of these fractions?
An $a \times b \times c$ rectangular box is built from $a \cdot b \cdot c$ unit cubes. Each unit cube is colored red, green, or yellow. Each of the $a$ layers of size $1 \times b \times c$ parallel to the $(b \times c)$ faces of the box contains exactly $9$ red cubes, exactly $12$ green cubes, and some yellow cubes. Each of the $b$ layers of size $a \times 1 \times c$ parallel to the $(a \times c)$ faces of the box contains exactly $20$ green cubes, exactly $25$ yellow cubes, and some red cubes. Find the smallest possible volume of the box.