Modified Fibonacci Golden Nuggets
Consider the infinite polynomial series $A_G(x) = x G_1 + x^2 G_2 + x^3 G_3 + \cdots$, where $G_k$ is the $k$th term of the second order...
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John Rhodes
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August 15, 2023
Pythagorean Tiles
Let $(a, b, c)$ represent the three sides of a right angle triangle with integral length sides. It is possible to place four such triangles together to...
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John Rhodes
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August 15, 2023
Special Isosceles Triangles
Consider the isosceles triangle with base length, $b = 16$, and legs, $L = 17$. By using the Pythagorean theorem it can be seen that the height...
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John Rhodes
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August 15, 2023
Fibonacci Golden Nuggets
Consider the infinite polynomial series $A_F(x) = x F_1 + x^2 F_2 + x^3 F_3 + \dots$, where $F_k$ is the $k$th term in the Fibonacci sequence:...
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John Rhodes
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August 15, 2023
Singleton Difference
The positive integers, $x$, $y$, and $z$, are consecutive terms of an arithmetic progression. Given that $n$ is a positive integer, the equation, $x^2 – y^2 –...
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John Rhodes
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August 15, 2023
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...