A Modified Collatz Sequence
A modified Collatz sequence of integers is obtained from a starting value $a_1$ in the following way: $a_{n+1} = \, \,\, \frac {a_n} 3 \quad$ if $a_n$...
-
John Rhodes
-
August 15, 2023
Primitive Triangles
Consider the triangles with integer sides $a$, $b$ and $c$ with $a \le b \le c$. An integer sided triangle $(a,b,c)$ is called primitive if $\gcd(a, b,...
-
John Rhodes
-
August 15, 2023
Balanced Sculptures
Let us define a balanced sculpture of order $n$ as follows: A polyominoAn arrangement of identical squares connected through shared edges; holes are allowed. made up of...
-
John Rhodes
-
August 15, 2023
Divisibility Multipliers
For each integer $p \gt 1$ coprime to $10$ there is a positive divisibility multiplier $m \lt p$ which preserves divisibility by $p$ for the following function...
-
John Rhodes
-
August 15, 2023
Sum of Squares
Consider equations of the form: $a^2 + b^2 = N$, $0 \le a \le b$, $a$, $b$ and $N$ integer. For $N=65$ there are two solutions: $a=1$,...
-
John Rhodes
-
August 15, 2023
Modular Cubes, Part 2
For a positive number $n$, define $C(n)$ as the number of the integers $x$, for which $1 \lt x \lt n$ and$x^3 \equiv 1 \bmod n$. When...
-
John Rhodes
-
August 15, 2023
Modular Cubes, Part 1
For a positive number $n$, define $S(n)$ as the sum of the integers $x$, for which $1 \lt x \lt n$ and$x^3 \equiv 1 \bmod n$. When...
-
John Rhodes
-
August 15, 2023
Cutting Squares
A square piece of paper with integer dimensions $N \times N$ is placed with a corner at the origin and two of its sides along the $x$-...