Found problems: 1048
There are finitely many primes dividing the numbers $\{ a \cdot b^n + c\cdot d^n : n=1, 2, 3,... \}$ where $a, b, c, d$ are positive integers. Prove that $b=d$.
Find all the positive integers $a,b,c$ such that $3ab= 2c^2$ and $a^3+b^3+c^3$ is the double of a prime number.
What is the largest positive integer that is not the sum of a positive integral multiple of 42 and a positive composite integer?
Prove that $n^4 + 4^{n}$ is composite for all values of $n$ greater than $1$.
Find the smallest odd prime $p$, such that there exist coprime positive integers $k$ and $\ell$ which satisfy
\[4k-3\ell=12\quad \text{ and }\quad \ell^2+\ell k +k^2\equiv 3\text{ }(\text{mod }p)\]
Let $p>2$ be a prime. Define a sequence $(Q_{n}(x))$ of polynomials such that $Q_{0}(x)=1, Q_{1}(x)=x$ and $Q_{n+1}(x) =xQ_{n}(x) + nQ_{n-1}(x)$ for $n\geq 1.$ Prove that $Q_{p}(x)-x^p $ is divisible by $p$ for all integers $x.$
Let $a$ and $b$ be positive integers not divisible by $5$. A sequence of integers is constructed as follows: the first term is $5$, and every consequent term is obtained by multiplying its precedent by $a$ and adding $b$. (For example, if $a = 2$ and $b = 4$, the first three terms are $5,14,32$.) What is the maximum possible number of primes that can occur before encoutering the first composite term?
Let $p$ be an odd prime number.
a) Show that $p$ divides $n2^n + 1$ for infinitely many positive integers n.
b) Find all $n$ satisfy condition above when $p = 3$
Find all triples of primes $(p, q, r)$ such that $p^q=2021+r^3$.
Let $A$ be a finite subset of prime numbers and $a> 1$ be a positive integer. Show that the number of positive integers $m$ for which all prime divisors of $a^m-1$ are in $A$ is finite.
Let $x$ > $1$ be an integer. Prove that $x^5$ + $x$ + $1$ is divisible by at least two distinct prime numbers.
For prime number $p$, prove that there are integers $a$, $b$, $c$, $d$ such that for every integer $n$, the expression $n^4+1-\left( n^2+an+b \right) \left(n^2+cn+d \right)$ is a multiple of $p$.
Find all prime numbers $p$ and $q$ such that $2^2+p^2+q^2$ is also prime.
Please remember to hide your solution. (by using the hide tags of course.. I don't literally mean that you should hide it :ninja: )
A prime number $p \ge 5$ is given. Write $\frac13+\frac24+... +\frac{p -3}{p - 1}=\frac{a}{b}$ for natural numbers $a$ and $b$. Show that $p$ divides $a$.
Let $p_{n}$ again denote the $n$th prime number. Show that the infinite series \[\sum^{\infty}_{n=1}\frac{1}{p_{n}}\] diverges.
Prime numbers $a, b, c$ are bigger that $3$. Show that $(a - b)(b - c)(c - a)$ is divisible by $48$.
Write $102$ as the sum of the largest number of distinct primes.
Let $p$ is a prime and $p\equiv 2\pmod 3$. For $\forall a\in\mathbb Z$, if
$$p\mid \prod\limits_{i=1}^p(i^3-ai-1),$$then $a$ is called a "GuGu" number. How many "GuGu" numbers are there in the set $\{1,2,\cdots ,p\}?$
(We are allowed to discuss now. It is after 00:00 Feb 14 Beijing Time)
We arranged all the prime numbers in the ascending order: $p_1=2<p_2<p_3<\cdots$.
Also assume that $n_1<n_2<\cdots$ is a sequence of positive integers that for all $i=1,2,3,\cdots$ the equation $x^{n_i} \equiv 2 \pmod {p_i}$ has a solution for $x$.
Is there always a number $x$ that satisfies all the equations?
[i]Proposed by Mahyar Sefidgaran , Yahya Motevasel[/i]
The digits $ 1$, $ 2$, $ 3$, $ 4$, $ 5$, $ 6$, $ 7$, and $ 9$ are used to form four two-digit prime numbers, with each digit used exactly once. What is the sum of these four primes?
$ \text{(A)}\ 150 \qquad
\text{(B)}\ 160 \qquad
\text{(C)}\ 170 \qquad
\text{(D)}\ 180 \qquad
\text{(E)}\ 190$
Let $p \geq 3$ be a prime number. Prove that there is a permutation $(x_1,\ldots, x_{p-1})$ of $(1,2,\ldots,p-1)$ such that $x_1x_2 + x_2x_3 + \cdots + x_{p-2}x_{p-1} \equiv 2 \pmod p$.
Two distinct 2-digit prime numbers $p,q$ can be written one after the other in 2 different ways to form two 4-digit numbers. For example, 11 and 13 yield 1113 and 1311. If the two 4-digit numbers formed are both divisible by the average value of $p$ and $q$, find all possible pairs $\{p,q\}$.
Suppose $S = \{a_1, a_2,..., a_{15}\}$ is a set of $1 5$ distinct positive integers chosen from $2 , 3, ... , 2012$ such that every two of them are coprime. Prove that $S$ contains a prime number.
(Note: Two positive integers $m, n$ are coprime if their only common factor is 1)
Find all natural numbers $ n$ for which every natural number whose decimal representation has $ n \minus{} 1$ digits $ 1$ and one digit $ 7$ is prime.
For all prime $p>3$ with reminder $1$ or $3$ modulo $8$ prove that the number triples $(a,b,c), p=a^2+bc, 0<b<c<\sqrt{p}$ is odd.
[i]Proposed by Navid Safaie[/i]