Computational Mathematics

John Rhodes
August 15, 2023
Computational Mathematics

Möbius Function and Intervals

The Möbius function, denoted $\mu(n)$, is defined as: $\mu(n) = (-1)^{\omega(n)}$ if $n$ is squarefree (where $\omega(n)$ is the number of distinct prime factors of $n$) $\mu(n)...

John Rhodes
August 15, 2023
Computational Mathematics

A Weird Recurrence Relation

The function $f$ is defined for all positive integers as follows: $f(1)=1$ $f(3)=3$ $f(2n)=f(n)$ $f(4n + 1)=2f(2n + 1) – f(n)$ $f(4n + 3)=3f(2n + 1) –...

John Rhodes
August 15, 2023
Computational Mathematics

Permutation of 3-smooth Numbers

A $3$-smooth number is an integer which has no prime factor larger than $3$. For an integer $N$, we define $S(N)$ as the set of $3$-smooth numbers...

John Rhodes
August 15, 2023
Computational Mathematics

Let $f_n(k) = e^{k/n} – 1$, for all non-negative integers $k$. Remarkably, $f_{200}(6)+f_{200}(75)+f_{200}(89)+f_{200}(226)=\underline{3.1415926}44529\cdots\approx\pi$. In fact, it is the best approximation of $\pi$ of the form $f_n(a) +...

John Rhodes
August 15, 2023
Computational Mathematics

An Ant on the Move

On the Euclidean plane, an ant travels from point $A(0, 1)$ to point $B(d, 1)$ for an integer $d$. In each step, the ant at point $(x_0,...

John Rhodes
August 15, 2023
Computational Mathematics

Flipping Game

The flipping game is a two player game played on an $N$ by $N$ square board. Each square contains a disk with one side white and one...

John Rhodes
August 15, 2023
Computational Mathematics

Permutations of Project

Consider the alphabet $A$ made out of the letters of the word “$\text{project}$”: $A=\{\text c,\text e,\text j,\text o,\text p,\text r,\text t\}$. Let $T(n)$ be the number of...

John Rhodes
August 15, 2023
Computational Mathematics

A Polynomial Modulo the Square of a Prime

Let $f(n) = n^2 – 3n – 1$. Let $p$ be a prime. Let $R(p)$ be the smallest positive integer $n$ such that $f(n) \bmod p^2 =...