Found problems: 17
It is given that the number $4^{11}+1$ is divisible by some prime greater than $1000$. Determine this prime.
[i] Proposed by David Altizio [/i]
Prove that number $2^{2008}\cdot2^{2010}+5^{2012}$ is not prime
Let $P$ be a function defined by $P(t)=a^t+b^t$, where $a$ and $b$ are complex numbers. If $P(1)=7$ and $P(3)=28$, compute $P(2)$.
[i] Proposed by Justin Stevens [/i]
Determine all positive integers $a,b,c$ satisfying $a^{(b^c)}=(b^a)^c$
Find all integers $x$ and $y$ such that $2^x+1=y^2$
If ${si}(x) =- \int \limits_{x}^{\infty}\left(\frac{\sin t}{t}\right)dt; x>0$ then $$\int \limits_{e}^{e^2} \left(\frac{1}{x}\left(si\left(e^4x\right)-si\left(e^3x\right)\right)\right)\,dx=\int \limits_{3}^{e^4} \left(\frac{1}{x}\left(\operatorname{si}\left(e^2x\right)-si\left(ex\right)\right)\right)dx$$
Find all positive integer solution:
$$k^m+m^n=k^n+1$$
[I]Proposed by V. Frank, F. Petrov[/i]
Find all pairs of positive integers $(m,n)$ for which there exist relatively prime integers $a$ and $b$ greater than $1$ such that
$$\frac{a^m+b^m}{a^n+b^n}$$
is an integer.
For every positive integeer $n>1$, let $k(n)$ the largest positive integer $k$ such that there exists a positive integer $m$ such that $n = m^k$.
Find $$lim_{n \rightarrow \infty} \frac{\sum_{j=2}^{j=n+1}{k(j)}}{n}$$
The value of $\left(256\right)^{.16}\left(256\right)^{.09}$ is:
$ \textbf{(A)}\ 4 \qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 64\qquad\textbf{(D)}\ 256.25\qquad\textbf{(E)}\ -16$
Find all triples of positive integers $(x, y, z)$ that satisfy the equation
$$2020^x + 2^y = 2024^z.$$
[i]Proposed by Ognjen Tešić, Serbia[/i]
Find the smallest positive integer $m$ such that $5m$ is an exact 5th power, $6m$ is an exact 6th power, and $7m$ is an exact 7th power.
Determine all positive integers $a$ for which $a^a$ is divisible by $20^{19}$.
Find the smallest prime number that can not be written in the form $\left| 2^a-3^b \right|$ with non-negative integers $a,b$.
What is the value of $$2^{\left(0^{\left(1^9\right)}\right)}+\left(\left(2^0\right)^1\right)^9\,?$$
$\textbf{(A) } 0 \qquad\textbf{(B) } 1 \qquad\textbf{(C) } 2 \qquad\textbf{(D) } 3 \qquad\textbf{(E) } 4$
Find all the positive integers $n$ so that $n$ has the same number of digits as its number of different prime factors and the sum of these different prime factors is equal to the sum of exponents of all these primes in factorization of $n$.
What is the units digit of $2022^{2023} + 2023^{2022}$?
$\textbf{(A)}~7\qquad\textbf{(B)}~1\qquad\textbf{(C)}~3\qquad\textbf{(D)}~5\qquad\textbf{(E)}~9$