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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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