Panaitopol Primes
A prime number $p$ is called a Panaitopol prime if $p = \dfrac{x^4 – y^4}{x^3 + y^3}$ for some positive integers $x$ and $y$. Find how many...
-
John Rhodes
-
August 15, 2023
Digital Signature
How many integers $0 \le n \lt 10^{18}$ have the property that the sum of the digits of $n$ equals the sum of digits of $137n$? To...
-
John Rhodes
-
August 15, 2023
Eulerian Cycles
Let $C(x, y)$ be a circle passing through the points $(x, y)$, $(x, y + 1)$, $(x + 1, y)$ and $(x + 1, y + 1)$....
-
John Rhodes
-
August 15, 2023
An Enormous Factorial
For any prime $p$ the number $N(p, q)$ is defined by $N(p, q) = \sum_{n = 0}^q T_n \cdot p^n$ with $T_n$ generated by the following random...
-
John Rhodes
-
August 15, 2023
Quadtree Encoding (a Simple Compression Algorithm)
The quadtree encoding allows us to describe a $2^N \times 2^N$ black and white image as a sequence of bits (0 and 1). Those sequences are to...
-
John Rhodes
-
August 15, 2023
Scoring Probabilities
Barbara is a mathematician and a basketball player. She has found that the probability of scoring a point when shooting from a distance $x$ is exactly $(1...
-
John Rhodes
-
August 15, 2023
Pythagorean Odds
Albert chooses a positive integer $k$, then two real numbers $a, b$ are randomly chosen in the interval $[0,1]$ with uniform distribution. The square root of the...
-
John Rhodes
-
August 15, 2023
Integer Sided Triangles with Integral Area/perimeter Ratio
Consider the triangle with sides $6$, $8$, and $10$. It can be seen that the perimeter and the area are both equal to $24$. So the area/perimeter...