Large Repunit Factors
A number consisting entirely of ones is called a repunit. We shall define $R(k)$ to be a repunit of length $k$. For example, $R(10) = 1111111111 =...
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John Rhodes
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August 15, 2023
Prime Cube Partnership
There are some prime values, $p$, for which there exists a positive integer, $n$, such that the expression $n^3 + n^2p$ is a perfect cube. For example,...
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John Rhodes
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August 15, 2023
Composites with Prime Repunit Property
A number consisting entirely of ones is called a repunit. We shall define $R(k)$ to be a repunit of length $k$; for example, $R(6) = 111111$. Given...
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John Rhodes
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August 15, 2023
Repunit Divisibility
A number consisting entirely of ones is called a repunit. We shall define $R(k)$ to be a repunit of length $k$; for example, $R(6) = 111111$. Given...
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John Rhodes
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August 15, 2023
Hexagonal Tile Differences
A hexagonal tile with number $1$ is surrounded by a ring of six hexagonal tiles, starting at “12 o’clock” and numbering the tiles $2$ to $7$ in...
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John Rhodes
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August 15, 2023
ABC Hits
The radical of $n$, $\operatorname{rad}(n)$, is the product of distinct prime factors of $n$. For example, $504 = 2^3 \times 3^2 \times 7$, so $\operatorname{rad}(504) = 2...
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John Rhodes
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August 15, 2023
Cuboid Layers
The minimum number of cubes to cover every visible face on a cuboid measuring $3 \times 2 \times 1$ is twenty-two. If we then add a second...
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John Rhodes
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August 15, 2023
Palindromic Sums
The palindromic number $595$ is interesting because it can be written as the sum of consecutive squares: $6^2 + 7^2 + 8^2 + 9^2 + 10^2 +...