Identity, Maths Club of IISER Kolkata

Girls only love men as tall as 6' and above. Socrates, ca. 2023 It is undeniable that heights strongly influence our daily lives. Be it our heights, or the height of a mountain we scale, or the height of all problems - humans. Mathematics too hasn't been able to escape its clutches, with height functions being useful in several fields, including but not limited to - Diophantine Geometry, Automorphic forms and the Weil-Mordell theorem - something you should have heard before if you attend my talks. If you have attended school (or maybe you are a climate activist) - then try recalling the elementary school days when fractions were introduced. Albeit unknowingly, but we had as children classified fractions into proper and improper - based on whether the denominator was larger than the numerator or vice versa. Well, it seems mathematicians have stuck with this classification - giving us the crux of todays discussion - height of a rational number. Given a rational number $x=\frac mn...

... or How I Learnt to Stop Worrying and Accept Mathematical Rigour. In Measure Theory, among other things, we try to generalize the concept of 'volume' (or 'area' or 'length' according as which one you prefer). In other words, like some sets come with an inherent 'measure' associated with them (for example, intervals in $\mathbb R$ comes with a length and "well behaved" shapes come with an area or volume depending on how many dimensions you count), we want to assign to any general subset a measure of its own. Since any general subset may not look as structured as the other sets that have their own volume, ideally we are free to randomly associate any number we want to to the set. But, as in all of mathematics, we want these numbers to satisfy certain properties. So, we will impose on these numbers a few properties that are very intuitive and geometric, but to our utter shock, are inconsistent. These properties are so much in tune with our ever...

This post mainly concerns the age old question of classifying all right-angled triangles with integer sides. In other words, we wish to find all integer solutions for the equation \[ X^2 + Y^2 = Z^2. \tag{1} \] An observant reader may have noticed that if $(X, Y, Z)$ is a solution, then so is $(kX, kY, kZ)$ for any integer $k$, and vice versa. This means that we can discard all common factors of $X, Y, Z$, and focus on solving $(1)$, with the added condition $\gcd(X, Y, Z) = 1$. Such solutions are called primitive solutions . You may remember a parameterized expression for these solutions, but do you know how to derive it? This is what we will be discussing here. There are multiple ways of arriving at the parametrization, but the one we'll use traverses the poetic bridge between numbers and geometry (the two divine deities of mathematics) with such ease, that it definitely makes it the best among them all. Rational hunt There is another simplification we c...