Faulhaber’s Formulas
The sum of the $k$th powers of the first $n$ positive integers can be expressed as a polynomial of degree $k+1$ with rational coefficients, the Faulhaber’s Formulas:...
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John Rhodes
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August 15, 2023
Chromatic Conundrum
Let $F(r, c, n)$ be the number of ways to colour a rectangular grid with $r$ rows and $c$ columns using at most $n$ colours such that...
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John Rhodes
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August 15, 2023
Prime-Sum Numbers
Define function $P(n, k) = 1$ if $n$ can be written as the sum of $k$ prime numbers (with repetitions allowed), and $P(n, k) = 0$ otherwise....
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John Rhodes
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August 15, 2023
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...
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John Rhodes
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August 15, 2023
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...
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John Rhodes
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August 15, 2023
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....
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John Rhodes
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August 15, 2023
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...
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John Rhodes
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August 15, 2023
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,...