Computational Mathematics

John Rhodes
August 15, 2023
Computational Mathematics

Euler’s Totient Function Equals 13!

The first number $n$ for which $\phi(n)=13!$ is $6227180929$. Find the $150\,000$th such number. The Euler totient function $\phi(n)$ calculates the number of positive integers that are...

John Rhodes
August 15, 2023
Computational Mathematics

Tangents to an Ellipse

A definition for an ellipse is: Given a circle $c$ with centre $M$ and radius $r$ and a point $G$ such that $d(G,M) \lt r$, the locus...

John Rhodes
August 15, 2023
Computational Mathematics

Coresilience

We shall call a fraction that cannot be cancelled down a resilient fraction. Furthermore we shall define the resilience of a denominator, $R(d)$, to be the ratio...

John Rhodes
August 15, 2023
Computational Mathematics

Resilience

A positive fraction whose numerator is less than its denominator is called a proper fraction. For any denominator, $d$, there will be $d – 1$ proper fractions;...

John Rhodes
August 15, 2023
Computational Mathematics

Odd Triplets

Given the set $\{1,2,\dots,n\}$, we define $f(n, k)$ as the number of its $k$-element subsets with an odd sum of elements. For example, $f(5,3) = 4$, since...

John Rhodes
August 15, 2023
Computational Mathematics

Perfection Quotients

For a positive integer $n$, let $\sigma(n)$ be the sum of all divisors of $n$. For example, $\sigma(6) = 1 + 2 + 3 + 6 =...

John Rhodes
August 15, 2023
Computational Mathematics

Top Dice

There are $1111$ ways in which five $6$-sided dice (sides numbered $1$ to $6$) can be rolled so that the top three sum to $15$. Some examples...

John Rhodes
August 15, 2023
Computational Mathematics

Twenty-two Foolish Primes

A set of disks numbered $1$ through $100$ are placed in a line in random order. What is the probability that we have a partial derangement such...