Same Differences
Given the positive integers, $x$, $y$, and $z$, are consecutive terms of an arithmetic progression, the least value of the positive integer, $n$, for which the equation,...
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John Rhodes
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August 15, 2023
Prime Pair Connection
Consider the consecutive primes $p_1 = 19$ and $p_2 = 23$. It can be verified that $1219$ is the smallest number such that the last digits are...
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John Rhodes
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August 15, 2023
Repunit Nonfactors
A number consisting entirely of ones is called a repunit. We shall define $R(k)$ to be a repunit of length $k$; for example, $R(6) = 111111$. Let...
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John Rhodes
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August 15, 2023
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...