# Tagged Questions

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### Almost perfect numbers

A positive integer $n$ is called almost perfect if the sum of its divisors smaller than $n$ is $n-1$. What are all almost perfect numbers $n$ such that some power $n^k$ is also almost perfect for at ...
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### Finding all primes $(p,q)$ for perfect squares.

Find all prime pairs $(p,q)$ such that $2p-1, 2q-1, 2pq-1$ are all perfect squares. Source: St.Petersburg Olympiad 2011 I have only found the pair $(5,5)$ so I am thinking that maybe a modulo $5$ ...
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### Curious number theory problem

$k,m,n\in\mathbb{N}$ satisfy $k^{m+n}=nm^n$. How can I show that $m=k$ and $n=k^k?$
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### $a^3+3a^2+a$ is never a perfect square.

Prove that no number of the form $a^3+3a^2+a$, for a positive integer $a$, is a perfect square. This problem was published in the Italian national competition (Cesenatico 1991). I've been ...
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### For any positive integers $a$ and $b$, the number $(36a+b)(a+36b)$ can never be a power of $2$.

APMO 1998: Show that for any positive integers $a$ and $b$, the number $(36a+b)(a+36b)$ can never be a power of $2$. The solution I've read substitutes $a=2^Ap,b=2^Bq$ where $p$ and $q$ are ...
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### Math Olympiads: GCD of terms in a sequence equals GCD of terms in other sequence

Recently, someone asked for a proof of a problem from the Russian Mathematical Olympiad, 1995. Math Olympiads: GCD of terms in a sequence equals GCD of their indices. The problem was to show that if ...
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### Math Olympiads: GCD of terms in a sequence equals GCD of their indices.

The sequence $a_1 ,a_2 ,a_3 ,...$ of positive integers satisfies $\text{gcd}(a_i ,a_j ) = \text{gcd} (i, j)$ for $i \neq j$. Prove that $a_i = i$ for all $i$. Source: Russian Mathematical Olympiad, ...
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### Prove that $(2m+1)^2 - 4(2n+1)$ can never be a perfect square where m, n are integers

I could prove it hit and trial method. But I was thinking to come up with a general and a more 'mathematically' correct method, but I did not reach anywhere. Thanks a lot for any help.
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### Show the sum is equal to a product of six primes

On a set of math challenges, one of them is to prove that 145678+456781+567814+678145+781456+814567 is the product of six different primes. This sounds like number theory to me, but I have no ...
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### A generalization of IMO 1977 problem 2

Here is the IMO 1977 problem 2: In a finite sequence of real numbers the sum of any seven successive terms is negative, and the sum of any eleven successive terms is positive. Determine the ...
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### Prove that for every prime $p > 100$ and every integer $r$ $\exists a, b$ such that $p \mid a^2 + b^5 - r$

Prove that for every prime $p > 100$ and every integer $r$ $\exists a, b \in \mathbb{Z}$ such that $p \mid a^2 + b^5 - r$ Preferably without using Jacobi sums, I have already seen a solution using ...
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### chain of divisibility relation

Let $a$ and $b$ be positive integers such that $a | b^2, b^2 | a^3, a^3 | b^4, b^4 | a^5, \cdots$ Prove that $a = b$. My way is as follows: Let $A=v_p(a), B=v_p(b)$ be the exact power of a prime $p$ ...
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### Interesting continued fraction problem $|r_i-u_0/u_1|\le\frac1{k_ik_{i+1}}$
Let $u_0/u_1$ be a rational number in lowest terms, and write $u_0/u_1=\langle a_0, a_1,...,a_n\rangle$ in standard continued fraction notation. Show that if $0\le i<n$, then ...