Computational Mathematics

John Rhodes
August 15, 2023
Computational Mathematics

Geometric Progression with Maximum Sum

Let $S(k)$ be the sum of three or more distinct positive integers having the following properties: No value exceeds $k$. The values form a geometric progression. The...

John Rhodes
August 15, 2023
Computational Mathematics

Divisibility of Harmonic Number Denominators

The $n$th harmonic number $H_n$ is defined as the sum of the multiplicative inverses of the first $n$ positive integers, and can be written as a reduced...

John Rhodes
August 15, 2023
Computational Mathematics

Counting Primitive Pythagorean Triples

A Pythagorean triple consists of three positive integers $a, b$ and $c$ satisfying $a^2+b^2=c^2$. The triple is called primitive if $a, b$ and $c$ are relatively prime....

John Rhodes
August 15, 2023
Computational Mathematics

Odd Elimination

Start from an ordered list of all integers from $1$ to $n$. Going from left to right, remove the first number and every other number afterward until...

John Rhodes
August 15, 2023
Computational Mathematics

Maximum Quadrilaterals

Consider a positive integer sequence $S = (s_1, s_2, \dots, s_n)$. Let $f(S)$ be the perimeter of the maximum-area quadrilateral whose side lengths are $4$ elements $(s_i,...

John Rhodes
August 15, 2023
Computational Mathematics

Modulo Power Identity

Let $S(n)$ be the sum of all positive integers $m$ not exceeding $n$ having the following property: $a^{m + 4} \equiv a \pmod m$ for all integers...

John Rhodes
August 15, 2023
Computational Mathematics

Fractal Sequence

Consider the infinite integer sequence S starting with: $S = 1, 1, 2, 1, 3, 2, 4, 1, 5, 3, 6, 2, 7, 8, 4, 9, 1,...

John Rhodes
August 15, 2023
Computational Mathematics

Weak Queens

The classical eight queens puzzle is the well known problem of placing eight chess queens on an $8 \times 8$ chessboard so that no two queens threaten...