Alexandrian Integers
We shall call a positive integer $A$ an “Alexandrian integer”, if there exist integers $p, q, r$ such that: $$A = p \cdot q \cdot r$$ and...
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John Rhodes
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August 15, 2023
Heighway Dragon
Let D0 be the two-letter string “Fa”. For n≥1, derive Dn from Dn-1 by the string-rewriting rules: “a” → “aRbFR” “b” → “LFaLb” Thus, D0 = “Fa”,...
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John Rhodes
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August 15, 2023
Skew-cost Coding
Let A and B be bit strings (sequences of 0’s and 1’s). If A is equal to the leftmost length(A) bits of B, then A is said...
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John Rhodes
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August 15, 2023
Perfect Right-angled Triangles
Consider the right angled triangle with sides $a=7$, $b=24$ and $c=25$. The area of this triangle is $84$, which is divisible by the perfect numbers $6$ and...
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John Rhodes
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August 15, 2023
Balanced Numbers
A positive integer with $k$ (decimal) digits is called balanced if its first $\lceil k/2 \rceil$ digits sum to the same value as its last $\lceil k/2...
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John Rhodes
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August 15, 2023
The Primality of $2n^2 – 1$
Consider numbers $t(n)$ of the form $t(n) = 2n^2 – 1$ with $n \gt 1$. The first such numbers are $7, 17, 31, 49, 71, 97, 127$...
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John Rhodes
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August 15, 2023
Crack-free Walls
Consider the problem of building a wall out of $2 \times 1$ and $3 \times 1$ bricks ($\text{horizontal} \times \text{vertical}$ dimensions) such that, for extra strength, the...
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John Rhodes
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August 15, 2023
Totient Chains
Let $\phi$ be Euler’s totient function, i.e. for a natural number $n$, $\phi(n)$ is the number of $k$, $1 \le k \le n$, for which $\gcd(k, n)...