Found problems: 700
Find all positive integers $n$ such that $4^n+6^n+9^n$ is a square.
[i]David Yang, Alex Zhu.[/i]
Let $n$ be a positive integer.
Prove that every prime $p > 2$ that divides $(2-\sqrt{3})^n + (2+\sqrt{3})^n$ satisfy $p=1 (mod3)$
Call a pair of integers $a$ and $b$ square makers , if $ab+1$ is a perfect square.
Determine for which $n$ is it possible to divide the set $\{1,2, \dots , 2n\}$ into $n$ pairs of square makers.
Let $\{a_n\}$ be a sequence of natural numbers such that each prime number greater than $1402$ divides a member of that. Prove that the set of prime divisors of members of sequence $\{b_n\}$ which $b_n=a_1a_2...a_n-1$ , is infinite.
[i]Proposed by Navid Safaei[/i]
A number $n$ is [i]interesting[/i] if 2018 divides $d(n)$ (the number of positive divisors of $n$). Determine all positive integers $k$ such that there exists an infinite arithmetic progression with common difference $k$ whose terms are all interesting.
Find all positive integers $ n$ such that there exists a unique integer $ a$ such that $ 0\leq a < n!$ with the following property:
\[ n!\mid a^n \plus{} 1
\]
[i]Proposed by Carlos Caicedo, Colombia[/i]
Let $a_1\in\mathbb{Z}$, $a_2=a_1^2-a_1-1$, $\dots$ ,$a_{n+1}=a_n^2-a_n-1$. Prove that $a_{n+1}$ and $2n+1$ are coprime.
Let $n\geq 0$ be an integer and let $p \equiv 7 \pmod 8$ be a prime number. Prove that
\[ \sum^{p-1}_{k=1} \left \{ \frac {k^{2^n}}p - \frac 12 \right\} = \frac {p-1}2 . \]
[i]Călin Popescu[/i]
For positive integers $a, b, c$ (not necessarily distinct), suppose that $a+bc, b+ac, c+ab$ are all perfect squares. Show that $$a^2(b+c)+b^2(a+c)+c^2(a+b)+2abc$$ can be written as sum of two squares.
Find all pairs of integers $(m,n)$ such that $m^6 = n^{n+1} + n -1$.
Find all positive integers, such that there exist positive integers $a, b, c$, satisfying $\gcd(a, b, c)=1$ and $n=\gcd(ab+c, ac-b)=a+b+c$.
For any prime $p$, prove that there exists integer $x_0$ such that $p | (x^2_0 - x_0 + 3)$ $\Leftrightarrow$ there exists integer $y_0$ such that $p | (y^2_0 - y_0 + 25).$
Determine all positive integers $n$ for which there exists an integer $m$ such that ${2^{n}-1}$ is a divisor of ${m^{2}+9}$.
Find all positive integers $n$ such that the set $\{n,n+1,n+2,n+3,n+4,n+5\}$ can be partitioned into two subsets so that the product of the numbers in each subset is equal.
Find all prime numbers $p$ such that $5^p + 12^p$ is a perfect square
For a fixed positive integer $k$, prove that there exist infinitely many primes $p$ such that there is an integer $w$, where $w^2-1$ is not divisible by $p$, and the order of $w$ in modulus $p$ is the same as the order of $w$ in modulus $p^k$.
[i]Author: James Rickards[/i]
Find all positive integers $ n$ such that there exists a unique integer $ a$ such that $ 0\leq a < n!$ with the following property:
\[ n!\mid a^n \plus{} 1
\]
[i]Proposed by Carlos Caicedo, Colombia[/i]
Let $n \ge 3$ be an integer such that $4n+1$ is a prime number. Prove that $4n+1$ divides $n^{2n}-1$.
Determine all positive integers $n$ for which $\dfrac{n^2+1}{[\sqrt{n}]^2+2}$ is an integer. Here $[r]$ denotes the greatest integer less than or equal to $r$.
The sequence $a_0$, $a_1$, $a_2,$ $\ldots$ is defined as follows: \[a_0=2, \qquad a_{k+1}=2a_k^2-1 \quad\text{for }k \geq 0.\] Prove that if an odd prime $p$ divides $a_n$, then $2^{n+3}$ divides $p^2-1$.
[hide="comment"]
Hi guys ,
Here is a nice problem:
Let be given a sequence $a_n$ such that $a_0=2$ and $a_{n+1}=2a_n^2-1$ . Show that if $p$ is an odd prime such that $p|a_n$ then we have $p^2\equiv 1\pmod{2^{n+3}}$
Here are some futher question proposed by me :Prove or disprove that :
1) $gcd(n,a_n)=1$
2) for every odd prime number $p$ we have $a_m\equiv \pm 1\pmod{p}$ where $m=\frac{p^2-1}{2^k}$ where $k=1$ or $2$
Thanks kiu si u
[i]Edited by Orl.[/i]
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If $d$ is a positive integer such that $d \mid 5+2022^{2022}$, show that $d=2x^2+2xy+3y^2$ for some $x, y \in \mathbb{Z}$ iff $d \equiv 3,7 \pmod {20}$.
Which natural numbers can be expressed as the difference of squares of two integers?
Let $(x_{n}) \ n\geq 1$ be a sequence of real numbers with $x_{1}=1$ satisfying $2x_{n+1}=3x_{n}+\sqrt{5x_{n}^{2}-4}$
a) Prove that the sequence consists only of natural numbers.
b) Check if there are terms of the sequence divisible by $2011$.
Kelvin the Frog and Alex the Kat are playing a game on an initially empty blackboard. Kelvin begins by writing a digit. Then, the players alternate inserting a digit anywhere into the number currently on the blackboard, including possibly a leading zero (e.g. $12$ can become $123$, $142$, $512$, $012$, etc.). Alex wins if the blackboard shows a perfect square at any time, and Kelvin's goal is prevent Alex from winning. Does Alex have a winning strategy?
Let $p$ be an odd prime congruent to 2 modulo 3. Prove that at most $p-1$ members of the set $\{m^2 - n^3 - 1 \mid 0 < m,\ n < p\}$ are divisible by $p$.