Diophantine Equation
Consider quadratic Diophantine equations of the form: $$x^2 – Dy^2 = 1$$ For example, when $D=13$, the minimal solution in $x$ is $649^2 – 13 \times 180^2...
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John Rhodes
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August 15, 2023
Convergents of $e$
The square root of $2$ can be written as an infinite continued fraction. $\sqrt{2} = 1 + \dfrac{1}{2 + \dfrac{1}{2 + \dfrac{1}{2 + \dfrac{1}{2 + …}}}}$ The...
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John Rhodes
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August 15, 2023
Odd Period Square Roots
All square roots are periodic when written as continued fractions and can be written in the form: $\displaystyle \quad \quad \sqrt{N}=a_0+\frac 1 {a_1+\frac 1 {a_2+ \frac 1...
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John Rhodes
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August 15, 2023
Powerful Digit Counts
The $5$-digit number, $16807=7^5$, is also a fifth power. Similarly, the $9$-digit number, $134217728=8^9$, is a ninth power. How many $n$-digit positive integers exist which are also...
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John Rhodes
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August 15, 2023
Cubic Permutations
The cube, $41063625$ ($345^3$), can be permuted to produce two other cubes: $56623104$ ($384^3$) and $66430125$ ($405^3$). In fact, $41063625$ is the smallest cube which has exactly...
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John Rhodes
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August 15, 2023
Cyclical Figurate Numbers
Triangle, square, pentagonal, hexagonal, heptagonal, and octagonal numbers are all figurate (polygonal) numbers and are generated by the following formulae: Triangle $P_{3,n}=n(n+1)/2$ $1, 3, 6,...
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John Rhodes
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August 15, 2023
Prime Pair Sets
The primes $3$, $7$, $109$, and $673$, are quite remarkable. By taking any two primes and concatenating them in any order the result will always be prime....
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John Rhodes
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August 15, 2023
XOR Decryption
Each character on a computer is assigned a unique code and the preferred standard is ASCII (American Standard Code for Information Interchange). For example, uppercase A =...