Generalization of Pythagoras equation

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About 3 years ago I discussed following two Diophantine equations of degree 2:

In this post, we will see a slight generalization of the result involving Pythagorean triplets. Unlike Pythagoras equation, $x^2+y^2-z^2=0$, we will work with a little bit more general equation, namely: $ax^2+by^2+cz^2=0$, where $a,b,c\in \mathbb{Z}$. For proofs, one can refer to section 5.5 of Niven-Zuckerman-Montgomery’s An introduction to the theory of numbers.

Theorem: Let $a,b,c\in \mathbb{Z}$ be non-zero integers such that the product is square free. Then $ax^2+by^2+cz^2=0$ have a non-trivial solution in integers if and only if $a,b,c$ do not have same sign, and that $-bc, -ac, -ab$ are quadratic residues modulo $a,b,c$ respectively.

In fact, this result helps us determine the existence of a non-trivial solution of any degree 2 homogeneous equation in three variables, $f(X,Y,Z)=\alpha_1 X^2 +\alpha_2Y^2+\alpha_3Z^2+\alpha_4XY+\alpha_5YZ+\alpha_6ZX$ due to the following lemma:

Lemma: There exists a sequence of changes of variables (linear transformations) so that $f(X,Y,Z)$ can be written as an equation of the form $g(x,y,z)=ax^2+by^2+cz^2$ with $\gcd(a,b,c)=1$.

Now let’s consider the example. Let $f(x,y,z)=3x^2+5y^2+7z^2+9xy+11yz+13zx$, and we want to determine whether this $f(x,y,z)=0$ has a non-trivial solution. Firstly, we will do change of variables:

$\displaystyle{f(x,y,z)=3\left(x+\frac{3}{2}y +\frac{13}{6}z\right)^2 - \frac{7}{4}y^2 - \frac{85}{12}z^2 - \frac{17}{2}yz = g(x',y',z')}$

where $x' = x+\frac{3}{2}y +\frac{13}{6}z$, $y'=y$ and $z'=z$. Thus

$\displaystyle{12g(x',y',z')=36x'^2 - 21 y'^2 - 85z'^2 - 102y'z' = 36x'^2 - 21\left(y'+\frac{17}{7}z'\right)^2+\frac{272}{7}z'^2=h(x'',y'',z'')}$

where $x'' = x'$,$y'' = y'+\frac{17}{7}z'$ and $z''=z'$. Thus

$\displaystyle{7h(x''',y'',z'') = 252x''^2 - 147y''^2+272z''^2=7(6x'')^2-3(7y'')^2 + 17(4z'')^2 = F(X,Y,Z)}$

where $X=6x''$, $Y=7y''$ and $Z=4z''$. Now we apply the theorem to $7X^2-3Y^2+17Z^2=0$. Since all the coefficients are prime numbers, we can use quadratic reciprocity to conclude that the given equation has non-trivial solution (only non trivial thing to note that $-7\times 17$ is quadratic residue mod -3, is same as $-7\times 17$ is quadratic residue mod 3).

Building Mathematics

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Let’s talk about the work of a mathematician. When I was young (before highschool), I used to believe that anyone capable of using mathematics is a mathematician. The reason behind this was that being a mathematician was not a job for people like Brahmagupta, Aryabhatta, Fermat, Ramanujan (the names I knew when I was young). So by that definition, even a shopkeeper was a mathematician. And hence I had no interest in becoming a mathematician.

Then, during highschool, I came to know about the mathematics olympiad and was fascinated by the “easy to state but difficult to solve” problems from geometry, combinatorics, arithmetic and algebra (thanks to AMTIVipul Naik and Sai Krishna Deep) . I practiced many problems in hope to appear for the exam once in my life. But that day never came (due to bad education system of my state) and I switched to physics, just because there was lot of hype about how interesting our nature is (thanks to Walter Lewin).

In senior school I realised that I can’t do physics, I simply don’t like the thought process behind physics (thanks to Feynman). And luckily, around the same time, came to know what mathematicians do (thanks to Uncle Paul). Mathematicians “create new maths”. They may contribute according to their capabilities, but no contribution is negligible. There are two kinds of mathematicians, one who define new objects (I call them problem creators) and others who simplify the existing theories by adding details (I call them problem solvers). You may wonder that while solving a problem one may create bigger maths problems, and vice versa, but I am talking about the general ideologies. What I am trying to express, is similar to what people want to say by telling that logic is a small branch of mathematics (whereas I love maths just for its logical arguments).

A few months before I had to join college (in 2014), I decided to become a mathematician. Hence I joined a research institute (clearly not the best one in my country, but my concern was just to be able to learn as much maths as possible).  Now I am learning lots of advanced (still old) maths (thanks to Sagar SrivastavaJyotiraditya Singh and my teachers) and trying to make a place for myself, to be able to call myself a mathematician some day.

I find all this very funny. When I was young, I used to think that anyone could become a mathematician and there was nothing special about it. But now I everyday have to prove myself to others so that they give me a chance to become a mathematician. Clearly, I am not a genius like all the people I named above (or even close to them) but I still want to create some new maths either in form of a solution to a problem or foundations of new theory and call myself a mathematician. I don’t want it to end up like my maths olympiad dream.