Totient Maximum
Euler’s totient function, $\phi(n)$ [sometimes called the phi function], is defined as the number of positive integers not exceeding $n$ which are relatively prime to $n$. For...
-
John Rhodes
-
August 15, 2023
Magic 5-gon Ring
Consider the following “magic” 3-gon ring, filled with the numbers 1 to 6, and each line adding to nine. Working clockwise, and starting from the group of...
-
John Rhodes
-
August 15, 2023
Maximum Path Sum II
By starting at the top of the triangle below and moving to adjacent numbers on the row below, the maximum total from top to bottom is 23....
-
John Rhodes
-
August 15, 2023
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...
-
John Rhodes
-
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...
-
John Rhodes
-
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...
-
John Rhodes
-
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...
-
John Rhodes
-
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...