If $x$, $y$, and $z$ are three positive integers such that
$$ x^n + y^n = z^n $$
and the exponent $n$ is also a positive integer, then $n \leq 2$.
In a previous issue issue of Parabola (Vol 29 No 2 p.2), I discussed the effect on time measurement of Einstein’s two postulates of Special Relativity.
It is fairly generally known, even amongst not very advanced students of mathematics, that in addition to the many ingenious constructions with straight edge and compasses which were discovered by the ancient Greeks, there were a number of similar construction problems which defied all their efforts...
Q1043. An equilateral triangle $APQ$ is drawn so that $P,Q$ are on the sides $BC$ and $DC$ of a square $ABCD,$ with $|AP| =|AQ|.$ Show that the perimeter of $APQ$ is less than the perimeter of the triangle $ABD$ (unless $P$ is at $B$ and $Q$ is at $D$).
Q1035. Find all positive integers $n$ and $m$ such that $n$ is a factor of $4m-1$ and $m$ is a factor of $4n-1$.
If a number of copies of a shape can be fitted together to form a larger copy of the same shape, we call the shape a “replicating tile”, or a “rep–tile” for short.