Computational Mathematics

John Rhodes
August 15, 2023
Computational Mathematics

Minimum Values of the Carmichael Function

The Carmichael function $\lambda(n)$ is defined as the smallest positive integer $m$ such that $a^m = 1$ modulo $n$ for all integers $a$ coprime with $n$. For...

John Rhodes
August 15, 2023
Computational Mathematics

Nanobots on Geodesics

Bob is a manufacturer of nanobots and wants to impress his customers by giving them a ball coloured by his new nanobots as a present. His nanobots...

John Rhodes
August 15, 2023
Computational Mathematics

Chinese Leftovers

Let $g(a, n, b, m)$ be the smallest non-negative solution $x$ to the system: $x = a \bmod n$ $x = b \bmod m$ if such a...

John Rhodes
August 15, 2023
Computational Mathematics

GCD of Divisors

Every divisor $d$ of a number $n$ has a complementary divisor $n/d$. Let $f(n)$ be the sum of the greatest common divisor of $d$ and $n/d$ over...

John Rhodes
August 15, 2023
Computational Mathematics

$10$-substrings

A $10$-substring of a number is a substring of its digits that sum to $10$. For example, the $10$-substrings of the number $3523014$ are: 3523014 3523014 3523014...

John Rhodes
August 15, 2023
Computational Mathematics

Constrained Sums

Let $S(n, k, b)$ represent the number of valid solutions to $x_1 + x_2 + \cdots + x_k \le n$, where $0 \le x_m \le b^m$ for...

John Rhodes
August 15, 2023
Computational Mathematics

Randomized Binary Search

A secret integer $t$ is selected at random within the range $1 \le t \le n$. The goal is to guess the value of $t$ by making...

John Rhodes
August 15, 2023
Computational Mathematics

Largest Prime Factors of Consecutive Numbers

Let $f(n)$ be the largest prime factor of $n$. Let $g(n) = f(n) + f(n + 1) + f(n + 2) + f(n + 3) + f(n...