1
vote
1answer
21 views

(Counting problem) very interesting Modular N algebraic eqs - for combinatorics experts

We have some attempt to numerically solve this math problem, which means that we like to count the number of independent solutions of this set of six of modular N algebraic equations: $$ (1) x_1 ...
2
votes
2answers
45 views

$2^n$ modulo n where n is odd always yields either an even or $1$

I'm attempting to do a pidgeonhole proof to prove that for some odd integer n, there is always a $2^k$ such that $2^k \mod(n) = 1$. I know that $2^n \mod(n)$ will always yield either an even number or ...
1
vote
1answer
46 views

Minimum number of rows in a mod 12 multiplication table

Minimum number of rows you would need to write out in a mod 12 multiplication table to guarantee you wrote out an element with an inverse? I would think this would be just one row as 1 is its own ...
0
votes
1answer
40 views

Combinatorics and Linear Alegbra

$$T = \{1011, 0112, 2101\} \subset Z_3^4$$ Is there any efficient way to find the span for set T other than checking all 27 possibility? If so, how to do it?
6
votes
1answer
86 views

Rooks on a labeled chessboard

An $n\times n$ chessboard is constructed such that the coordinate $(i, j)$ is labeled with $i+j \mod n$. Example for n = 6: The goal is place $n$ rooks in the chessboard such that none threaten ...
13
votes
2answers
338 views

When is $\binom{n}{k}$ divisible by $n$?

Is there any way of determining if $\binom{n}{k} \equiv 0\pmod{n}$. Note that I am aware of the case when $n =p$ a prime. Other than that there does not seem to be any sort of pattern (I checked up ...
5
votes
4answers
117 views

${{p-1}\choose{j}}\equiv(-1)^j \pmod p$ for prime $p$

Can anyone share a link to proof of this? $${{p-1}\choose{j}}\equiv(-1)^j(\text{mod}\ p)$$ for prime $p$.
3
votes
0answers
86 views

Equivalence classes of triplets satisfying $x^2+y^2+z^2=0$ over $\mathbb{F}_p$

The affirmative answer to this question illustrates that the equation $$x^2+y^2+z^2=0$$ has $p^2-1$ nontrivial solutions over $\mathbb{F}_p$ (solutions that are not $(0,0,0)$). If $(x,y,z)$ is a ...
1
vote
0answers
300 views

Binomial coefficient modulo prime power without generalized Lucas theorem

I've been working on this problem computing $n \choose r$ for large $n$ and $r$, modulo a composite. I could implement the generalized lucas theorem to handle the prime power case, but I want to ...
0
votes
0answers
72 views

Distribution of binary digits in moduli

Considering the (infinite) set of all positive integers that are a product of $2$ primes only, represented in binary $100...01$. Question: is the distribution of the proportion of $0,1$ digits ...
1
vote
2answers
192 views

Function with a Modular Inverse

For a combinatorics problem I have a function, $h(x)$ that is always divisible by five, but it is calculated in pieces, e.g. $h(1) = 43 + 7$. The final function that I need is $f(x) = (h(x) / 5) ...
2
votes
2answers
92 views

Counting the number of matrices which cause collision

Let $m,n \in \mathbb{N}$, and $q$ be a prime number. Let $\mathbf{A} \in \mathbb{Z}^{m \times n}_q$ be a matrix. In the following, assume that all additions and multiplications are performed modulo ...
0
votes
2answers
132 views

$k$ hands in $n$'s hair

Moderator Message: this question is from an ongoing competition. Define a prime $p$ as having $k$ hands in $n$'s hair if $p^k|n$ and $n|2^n+1$ . Does there exist an integer $n$ with $2012$ hands ...
2
votes
0answers
192 views

$a^{(b^c)} \mod m$ where $c$ can be very very large

I am trying to solve the following problem. I need to find the value of $$ a^{(b^x)} \bmod m $$ where $a,b$ are integers and $$ x = \pmatrix{n\\0}^2 + \pmatrix{n\\1}^2 + ... + \pmatrix{n\\n}^2 ...
1
vote
2answers
114 views

Solutions to $x_1+2x_2+3x_3+4x_4+5x_5+6x_6+7x_7+8x_8+9x_9+10x_{10}\equiv0\mod11$

How many solutions does the following equation have: $x_1+2x_2+3x_3+4x_4+5x_5+6x_6+7x_7+8x_8+9x_9+10x_{10}\equiv0\mod11$ where $x_{1...9} \in \{0,1,2,3,4\ ...\ 8,9\}$ and $x_{10}\in\{0,1,2,3,4\ ...
4
votes
1answer
197 views

Fermat's little theorem for $n=3$

for $N > 0$, I'm trying to show Fermat's little theorem, for $3$ using the orbit stabilizer theorem: $N^3 - N$ an element of $3\mathbb{Z}\ (3 \mod \mathbb{Z})$ Pf/ we can break it down into ...
0
votes
3answers
67 views

Prove how many distinct elements in the set $\{ax \pmod{m}:a\in\{0,…,m-1\}\}$

There are $\dfrac{m}{\gcd(m,x)}$ distinct elements in the set $\{ax \pmod{m}:a\in\{0,...,m-1\}\}$ I have only known these by using a computer to generate the number of distinct elements. But I am not ...
6
votes
4answers
903 views

calculating n choose k mod one million

I am working on a programming problem where I need to calculate 'n choose k'. I am using the relation formula $$ {n\choose k} = {n\choose k-1} \frac{n-k+1}{k} $$ so I don't have to calculate huge ...