Sequence of Points on a Hyperbola
Let $H$ be the hyperbola defined by the equation $12x^2 + 7xy – 12y^2 = 625$. Next, define $X$ as the point $(7, 1)$. It can be...
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John Rhodes
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August 15, 2023
Prime Factors of $n^{15}+1$
Numbers of the form $n^{15}+1$ are composite for every integer $n \gt 1$. For positive integers $n$ and $m$ let $s(n,m)$ be defined as the sum of...
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John Rhodes
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August 15, 2023
$2 \times 2$ Positive Integer Matrix
A positive integer matrix is a matrix whose elements are all positive integers. Some positive integer matrices can be expressed as a square of a positive integer...
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John Rhodes
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August 15, 2023
Look and Say Sequence
The look and say sequence goes 1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, … The sequence starts with 1 and all other members are obtained by...
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John Rhodes
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August 15, 2023
Factorisation Triples
Let $n$ be a positive integer. An integer triple $(a, b, c)$ is called a factorisation triple of $n$ if:$1 \leq a \leq b \leq c$$a \cdot...
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John Rhodes
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August 15, 2023
Reciprocal Cycles II
A unit fraction contains $1$ in the numerator. The decimal representation of the unit fractions with denominators $2$ to $10$ are given: \begin{align} 1/2 &= 0.5\\ 1/3...
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John Rhodes
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August 15, 2023
A Frog’s Trip
A row of $n$ squares contains a frog in the leftmost square. By successive jumps the frog goes to the rightmost square and then back to the...
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John Rhodes
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August 15, 2023
Titanic Sets
A set of lattice points $S$ is called a titanic set if there exists a line passing through exactly two points in $S$. An example of a...