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Schinzel's Theorem

Today, I'll translate the proof for a theorem by André Schinzel that gives a circle with $n$ integer coordinates (also known as Lattice Points) on its circumference, for any given $n$. The theorem was published in L'Enseignement Mathématique in 1958 and can be found here.

Area and Perimeter of Astroids

It is about time I revisited my first adventure Envelopes and Astroids. In the end of the post, I named the envelope of the line segments an Astroid. I will now find two basic properties of this shape: area and perimeter.

The Power of Generating Functions

In Pentagonal Number Theory, I touched on the topic of generating function but now I'll give examples of generating functions being used to find explicit solutions for recurrent relations. I think this was their primary purpose; Abraham de Moivre invented them when he tried to find the exact formula for …

Surprising Surds

Yesterday, I stumbled upon this very surprising identity while reading on nested radicals: \[ \sqrt[3]{2\pm\sqrt{5}} = \frac{1\pm\sqrt{5}}{2} \] While it is easy to prove the fact after seeing it, I will try to prove this from the perspective of someone who is not …

Euler's Pentagonal Number Theorem

Today, I'll prove Euler's Pentagonal Number Theorem and show how he used it to find recurrence formulae for the sum of \(n\)'s positive divisors and the partitions of \(n\). This post will be based on two papers I read last week: “An Observation on the Sums of Divisors” and …

Envelopes and Astroids

Envelopes kept cropping up in my doodles but I never gave them much attention. Up until the day I started drawing line segments of a constant length, say \(1\), from one side of a piece of paper to an adjacent side (See Fig 1). It appeared that these lines were …

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