Computational Mathematics

John Rhodes
August 15, 2023
Computational Mathematics

Triangles Containing the Origin II

Define:$x_n = (1248^n \bmod 32323) – 16161$$y_n = (8421^n \bmod 30103) – 15051$ $P_n = \{(x_1, y_1), (x_2, y_2), \dots, (x_n, y_n)\}$ For example, $P_8 = \{(-14913,...

John Rhodes
August 15, 2023
Computational Mathematics

Diophantine Reciprocals III

In the following equation $x$, $y$, and $n$ are positive integers. $$\dfrac{1}{x} + \dfrac{1}{y} = \dfrac{1}{n}$$ For a limit $L$ we define $F(L)$ as the number of...

John Rhodes
August 15, 2023
Computational Mathematics

Lattice Quadrilaterals

A simple quadrilateral is a polygon that has four distinct vertices, has no straight angles and does not self-intersect. Let $Q(m, n)$ be the number of simple...

John Rhodes
August 15, 2023
Computational Mathematics

Long Products

Define $F(m,n)$ as the number of $n$-tuples of positive integers for which the product of the elements doesn’t exceed $m$. $F(10, 10) = 571$. $F(10^6, 10^6) \bmod...

John Rhodes
August 15, 2023
Computational Mathematics

Modular Inverses

Consider the number $15$. There are eight positive numbers less than $15$ which are coprime to $15$: $1, 2, 4, 7, 8, 11, 13, 14$. The modular...

John Rhodes
August 15, 2023
Computational Mathematics

Hypocycloid and Lattice Points

A hypocycloid is the curve drawn by a point on a small circle rolling inside a larger circle. The parametric equations of a hypocycloid centered at the...

John Rhodes
August 15, 2023
Computational Mathematics

Chocolate Covered Candy

Phil the confectioner is making a new batch of chocolate covered candy. Each candy centre is shaped like an ellipsoid of revolution defined by the equation: $b^2...

John Rhodes
August 15, 2023
Computational Mathematics

Average Least Common Multiple

The function $\operatorname{\mathbf{lcm}}(a,b)$ denotes the least common multiple of $a$ and $b$. Let $A(n)$ be the average of the values of $\operatorname{lcm}(n,i)$ for $1 \le i \le...