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...
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John Rhodes
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August 15, 2023
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,...
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John Rhodes
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August 15, 2023
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...
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John Rhodes
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August 15, 2023
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...
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John Rhodes
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August 15, 2023
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...
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John Rhodes
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August 15, 2023
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...
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John Rhodes
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August 15, 2023
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...
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John Rhodes
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August 15, 2023
$10$-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...