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The trick is to simplify as much as possible. Have you noticed that the $n$ never appears alone in your question, but only in the context of $p(n)$, so why don't you just write $p$ for $p(n)$ ? Then, …

The point is that you use this fact from number theory: $p$ and $q$ are relatively prime $\Longrightarrow$ $p^2$ and $q^2$ are relatively prime. While this is not hard, this is not trivial either. Y …

Proved by grobber (Alexandru Chirvasitu) on AoPS: http://www.artofproblemsolving.com/Forum/viewtopic.php?f=61&t=157732 EDIT: Also, http://www.artofproblemsolving.com/Forum/viewtopic.php?f=38&t=335001 …

We have $1\equiv u\left( p-r\right) \equiv u\left( -r\right) =-ur\operatorname{mod}p$, so that $p\mid1+ur=ur+1$. Hence, $\left\lceil \dfrac{ur}{p}\right\rceil =\dfrac{ur+1}{p}$. We need to prove t …

Persi Diaconis gave a talk about probabilities of carries at MIT about half a year ago. Unfortunately I was so busy trying to spot symmetric functions in the talk that most of it drifted past my mind, …

EDIT: Okay, wrong. Sorry for spamming. This looks just too easy, so I guess I'm doing something stupid. I'll prove that (a) whenever a subfield $K$ of $\mathbb{C}$ satisfies $\beta\in K\left[\alpha …

No, such $c_x$ don't exist. Even if you replace $\lbrace 0,\pm 1,\pm 2\rbrace$ by $\mathbb Q$, this won't change. In fact, if they would exist, then, using the relation $\displaystyle \sin\frac{2\pi x …

The answer is Yes. The only reason for the appearance of formally real fields and $\mathbb R$ in the question is to rule out roots of unity; in fact, we have the following (cleaner and more general) s …

First of all, let's formalize the definition: Let $\lambda$ be a partition. For every integer $i \geq 1$, let $m_i\left(\lambda\right)$ be the number of appearances of $i$ in $\lambda$. Then your fanc …

To your first question: No, it is false. For $n = 256 - 13$, the number $a_n$ has $\nu_2\left(a_n\right) = 7 < 8 = \nu_2 \left(n+13\right)$. HOWEVER, it is almost correct: namely, it is correct whene …

If you're in a hurry scroll down until the questions: First the known part: A sequence $\left(b_1,b_2,b_3,...\right)$ of integers will be called a ghost-Witt sequence if there exists a sequence $\le …

Solved. The key is that the set of polynomials $P$ for which there exist polynomials $P_d\in\mathbb{Z}\left[\Xi\right]$ for all divisors $d$ of $n$ such that $\displaystyle P=\sum_{d\mid n}dP_d^{n/d}$ …

This seems very much like homework to me, so I'll be brief. I assume that your $Z_p$ denotes the field with $p$ elements; I will call it $\mathbb{F}_p$ henceforth (lest it be confused with the ring $\ …

Here is the answer I hinted at in a comment, in real detail. Took me a while, but I had no idea how tiresome such arguments are to expose... Yes, it is true: see Corollary 6 (b) below. The proof reli …

This is merely a variation of your own proof, Matt, but I believe it makes things clearer. The first step is to define $c:=1-a-b$. Then, your identity takes the form $\dfrac{\Gamma\left(b\right)\Gam …