Articles of discrete mathematics

Decomposition by subtraction

In how many ways one can decompose an integer $n$ to smaller integers at least 3? for example 13 has the following decompositions: \begin{gather*} 13\\ 3,10\\ 4,9\\ 5,8\\ 6,7\\ 3,3,7\\ 3,4,6\\ 3,5,5\\ 3,3,3,4\\ 4,4,5\\ \end{gather*} Points and hints are welcome.

Is there a number that is palindromal in both base 2 and base 3?

I manually checked the first 20 base 2 palindromes and I did not find any base 3 palindromes among them. Is there any definite way of determining this? What about other bases?

How can we draw $14$ squares to obtain an $8 \times 8$ table divided into $64$ unit squares?

How can we draw $14$ squares to obtain an $8\times8$ table divided into $64$ unit squares? Notes: -The squares to be drawn can be of any size. -There will be no drawings outside the table.

How many possibilities to arrange a rope of length $N$ between two points

Consider a $n$-dimensional lattice with $M \times M \times … \times M$ ($n$ times) discrete grid cells ($n,M$ are natural numbers). Given two arbitrary cells at position $\vec{i}=(i_1,i_2,…,i_n)$ and $\vec{j}=(j_1,j_2,…,j_n)$ that a rope with discrete length $N$ interlinks (length is defined as the number of grid cells which can be filled in this lattice). A […]

Number of ways distribute 12 identical action figures to 5 children

Need a little help with this problem. Use generating functions to determine the number of different ways 12 identical action figures can be given to five children so that each child receives at most three action figures. So far I have that we are looking for the coefficient of $x^{12}$ and the generating function is […]

Prove that every number between two factors of primes is composite.

I am looking for some help with this problem: Let $p_1,p_2,\dots,p_{n+1}$ be the first $n+1$ primes in order. Prove that every number between $p_1p_2p_3\dots p_n+1$ and $p_1p_2p_3\dots p_n + p_{n+1}-1$ is composite (inclusive of the second term). How does this show that there are gaps of arbitrary length in the sequence of primes? I know […]

Why these two problems lead to same answers?

Suppose these two problems: Problem 1: $$\min_{X,P} \quad\max_{1\leq l\leq L-1} \quad {|\sum_{1\leq i\leq N_p}^{N_p}x_ie^{\frac{2\pi l}{N}p_i}| \over {\sum_{i=1}^{N_p} x_i^2}} \quad \equiv \quad \min_{X,P} \quad\max_{1\leq l\leq L-1} \quad {\sum_{1\leq i,j\leq N_p}^{N_p}x_ix_je^{\frac{2\pi l}{N}(p_i-p_j)} \over {(\sum_{i=1}^{N_p} x_i^2})^2} $$ Problem 2: $$\min_{P} \quad\max_{1\leq l\leq L-1} \quad |\sum_{1\leq i\leq N_p}^{N_p}e^{\frac{2\pi l}{N}p_i}| \quad \equiv \quad \min_{P} \quad\max_{1\leq l\leq L-1} \quad \sum_{1\leq i,j\leq […]

Prove that $C_n < 4n^2$ for all n greater than or equal to 1

$C_1 = 0$, $C_n = C_{\lfloor n/2\rfloor} + n^2$ for all $n \ge 1$ Prove that $C_n < 4n^2$ for all $n \ge 1$. I don’t know how to even approach this. I remember something about inductive proofs…but i really don’t understand that, could you please explain that to me?

A naive example of discrete Fourier transformation

We know a discrete Fourier transformation with discrete $n$ and continuous $x_1,x_2$: $$ \sum_{n\in\mathbb{Z}} e^{-in(x_1-x_2)\frac{2\pi}{L}}=L\delta(x_1-x_2) $$ with Dirac delta function $\delta$. and a discrete Fourier transformation with discrete $n$ and discrete $x_1,x_2$: $$ \sum_{n\in\mathbb{Z}} e^{-in(x_1-x_2)\frac{2\pi}{L}}=L\delta_{x_1-x_2,0} $$ with Kronecker delta function $\delta$. Question: what is the discrete Fourier transformation with discrete $n$ and discrete $x_1,x_2$ below: […]

Existence of uncountable set of uncountable disjoint subsets of uncountable set

“Can you to any uncountably infinite set $M$ find an uncountably infinite family $F$ consisting of pairwise disjoint uncountably infinite subsets of $M$?” Intuitively, I feel like it should be possible for the real numbers at least: you simply split the real numbers into two intervals, and since there are uncountably many points where you […]