# Tagged Questions

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### Simplifying a sum in terms of divisor function Cauchy products

I'm trying to simplify this combinatorial looking sum: $$\sum_{ax+by=n}_{(a,x,b,y)\in \mathbb{N^4}}\max{\{a,b\}}$$ In terms of possibly some scaled divisor functions plus a Cauchy product/convolution ...
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### What is the function like when its Mobius inversion is $\sum_{w|r, (w,t)=1}\mu(w)q^{r/w}$?

Everyone, I am now reading a paper named The Irreducible Factors of $(cx+d)x^{q^m}-(ax+b)$ over $GF(q)$, http://qjmath.oxfordjournals.org/content/14/1/61.extract. And I’m confused with one of its ...
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### Number of partitions whose blocks form arithmetic progressions

As is known, the set $\{1,\ldots,n\}$ has $2^n$ many subsets and $B_n$ (the $n$th Bell number) many partitions, where clearly $B_n<2^{2^n}$ and it is actually known that $B_n<n^n$ for large $n$. ...
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### A lower bound on the number of matrices whose image contains all multiples of $p^e$

Let $0\leq e<e^\prime$ be integers. Now suppose $N$ is the number of $n\times n$ matrices over the ring $R:=\mathbb{Z}/p^{e^\prime}\mathbb{Z}$ (where $p$ is prime) such that ...
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### Erdös-Fuchs Theorem for multivariate linear forms

Let $A$ be an infinite set of positive integers, and denote by $r(n)$ the number of solutions to the equation $a+a'=n$, with $a,\, a' \in A$. It is not very difficult to show that if $r(n) > 0$ ...
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### Linear independence of the square roots over Q

Does there exist a real number $a$ such that the numbers $\sqrt{n^2 + a^2}$ (for all natural $n$) are linearly independent over the field of rational numbers? It is evident that $a$ cannot be ...
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### Minimize $td-2\sum_{i=0}^{t-1} w_k(i)$ where $w_k(i)$ is the sum of the base-$k$ digits of $i$

Let $K_k^n$ denote the $n$-fold cartesian product of the complete graph on $k$ vertices, and let $[R,T]$ be the edge cut, consisting of the edges between complementary vertex sets $R$ and $T$. I ...
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### summation of products of combinatorials

For any natural number $N$ and $0\le n\le N$ define $$f(n) = f(n,N) = \frac{1}{(N+1)!} \sum_{\substack{{S\subset \{1,\ldots,N\}} \\ {|S|=n}}} \prod_{s\in S} s.$$ (The empty product is interpreted ...
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### Known results on cyclic difference sets

Is there any infinite family of $v$ for which all the $(v,k,\lambda)$-cyclic difference sets with $k-\lambda$ a prime power coprime to $v$ have been determined? A subset $D=\{a_1,\ldots,a_k\}$ of ...
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### Estimate the sum $\sum_{k=1}^n \frac{2^k}{k}$ [closed]

I am concerned with the following sum $$\sum_{k=1}^n \frac{2^k}{k}_{\displaystyle ~.}$$ It seems that sum must be computed by someone, but I do not know. By the way, I can figure out a power ...
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### Distribution of the evaluation (at a non-trivial root of -1) of polynomials with small coefficients

When we studied some cryptographic protocol, we came accross the following problem, which seems linked to the uniformity of the residues of small multiplicative subgroups of $\mathbb{F}_q$. Problem ...
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### Sets whose elements are mutually “weakly” coprime?

Fix $n$ and $k$. I want a set $S\subseteq\{1,\ldots,n\}$ with the property that for every $x\in S$, $$\mathrm{gcd}\bigg(x,\prod_{y\in S\setminus\{x\}}y\bigg)<\frac{x}{k}.$$ How small should a ...
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### A partition problem

This question is motivated by an old math contest problem, and is a generalization of the original problem. I will write out the original problem as motivation. Let us say that $n$ is $p$-Savage, for ...
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### Quadratic transformation of hypergeometric function 2F1

I want to know whether there is some transformation between $_2F_1(a,b;c;x)$ and $_2F_1(a',b';c';x(1-x))$. Here is an example called the Kummer quadratic transformation, which may be known to most of ...
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### Multivariate generating function

I am investigating the perturbation of the Jordan canonical form. In my work I must calculate the number of ways to factor $p^ {n-k} q^k$ where $p$ and $q$ are distinct primes ...
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### Avoiding multiples of $p$

Let $p$ be a prime number and $P=\{1,2,...,p-1\}$ In how many ways we can sum all the elements of $P$ in such a way that we will reach a multiple of $p$ only when we sum the last summand? For ...
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### On the vanishing of the generalized von Mangoldt function $\Lambda_k(n)$ when $n$ has more than $k$ prime factors

It is a well-known fact that the generalized von Mangoldt function, defined by $$\displaystyle \Lambda_k(n) = \sum_{d | n} \mu(d) \left(\log \frac{n}{d}\right)^k$$ vanishes whenever $n$ has more ...
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### A Question on 1, 2 ,3 Conjecture

1, 2, 3 conjecture is well-known: If $G$ is a simple graph which is not $K_2$ then one can assign a number among $1, 2, 3$ to every edge such that if we label each vertex with the sum of the numbers ...
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### Combinatorial Technique Needed

The following problem is likely too special for MO. However I have no clue how to deal with it, so I'll just try. Nevertheless it is a combinatorial problem and a discussion about general methods in ...
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### special values of L-functions cohen-lenstra heuristic

I found some lecture notes on links between number theory and random permutations. It was difficult to follow: The notes start with an interesting fact, whose proof I've asked on Math.StackExchange: ...
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### Infinitely many $N$ such that $\langle p\rangle=\langle q\rangle$ mod $N$

Suppose that $p,q>1$ are two relatively prime integers. Are there infinitely many positive integers $N$ such that $N$ is relatively prime to $p$ and $q$; there exists positive integers $k,l$ such ...
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### A Product Related to Unrestricted Partitions

Start with the product for unrestricted partitions: (1+x+x$^2$+...)(1+x$^2$+x$^4$+...)(1+x$^3$+x$^6$+...)... Now replace some of the plus signs with minus signs and expand the product into a ...
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### 2-adic Logarithm and Resistance of n-dimensional Cube

Resistance across opposite vertices of n-dimensional cube with each edge at one ohm resistance is $$R_n=\sum_{k=0}^{n-1}\frac1{(n-k){n\choose k}}=\frac1n\sum_{k=1}^{n}\frac1{{n-1\choose k}}.$$ The ...
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### Pythagorean triples related to non-isometric equidistant plane quadruples

QUESTION   Do there exist integers   $u\ x\ A\ B$   such that   $x\ne 0$,   and the following two equalities hold: $x^2 + (x-u)^2\ =\ A^2$ $x^2 + (x+u)^2\ =\ B^2$ ? ...
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### Periodic tilings of the plane with fundamental domain given by $k$ squares of prescribed size

Given $k$ strictly positive real numbers $l_1,\dots,l_k$, can one decide the existence of a periodic tiling of the plane whose fundamental domain is the union of $k$ squares of length ...
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### How to explain the picturesque patterns in François Brunault's matrix?

How to explain the patterns in the matrix defined in François Brunault's answer to the question Freeness of a Z[x] module depicted below? -- Choosing colors according to the highest power of 2 which ...
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### Upper Limit on the Central Binomial Coefficient

What is the tightest upper bound we can establish on the central binomial coefficients $2n \choose n$ ? I just tried to proceed a bit, like this: $n! > n^{\frac{n}{2}}$ for all $n>2$. ...
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### Discriminants of indefinite integral binary quadratic forms admitting 3 or 6 torsion.

Are there any results known about the discriminants of indefinite integral binary quadratic forms admitting automorphisms of order 3 or 6? It seems reasonable to expect that any permissible ...
### Independence of an interval and a product set in $\mathbb Z/L\mathbb Z$.
I have $k$ distinct prime numbers $\ell_1 < \dots <\ell_k$, and for each $i=1,\dots,k$, a subset $A_i$ of $\mathbb Z / \ell_i \mathbb Z$. Let $L=\ell_1 \dots \ell_k$. Now using the chinese ...
Definition: Let $r(m)$ denote the residue class $r+m\mathbb{Z}$, where $0 \leq r < m$. Given disjoint residue classes $r_1(m_1)$ and $r_2(m_2)$, let the class transposition ...