Nerd sniping

Last week a friend of mine, Felipe, left written the following equation in the blackboard of the coffee room: \[ \frac{a}{b+c} + \frac{b}{c+a} + \frac{c}{a+b} = 4,\]asking for a solution with positive integers.

It was a harmless sort of nerd sniping (as opposed to the harmful one), that he saw as a meme online. It looks innocent, but it is a rather difficult problem1. The idea was to post it to family WhatsApp groups, as revenge for the retarded math problems with fruits as variables that come up regularly. Finding the idea hilarious, I went on and posted it to my family’s WhatsApp group.

But… how does one solve it? I had inadvertently sniped myself. Trying a little bit, I managed to show that there is no integer solution for $a=b$, and therefore one couldn’t reduce the problem from a third-degree polynomial to a second degree one, so there was no easy way out. I also realised that any integer multiple of a solution is a valid solution, so one can fix $c=1$, look for a solution for rational $a,b$, and multiply this solution by the least common denominator to solve the original problem. Fixing then $c=1$ and rewriting $a,b$ as $x,y$ for clarity, we end up with the polynomial equation
\[ x^3-3x^2-3x^2y-3x + y^3-3y^2-3y^2x-3y-5xy + 1 = 0,\]which looks pretty much hopeless. To get some intuition I plotted the curve, obtaining this:

Well great. Now I was stuck. Off to Google then. Which immediately gave me this answer, a thorough explanation of where the problem came from and how to solve it. End of story? Not really. The answer depended on two magical steps2 that were neither easy nor explained. No no, I want to actually find the solution, not just reproduce it using stuff I don’t understand.

As the mathematician helpfully explains there, the core idea is the centuries-old chord and tangent technique: if draws a line through two rational points on the curve, or on a tangent point, this line will intersect the curve again on a rational point. So if we have a single rational solution to start with, we can just iterate away and produce new rational points.

Well, this I can understand and do myself, off to work then! I did everything on the programming language that’s all the rage with the kids nowadays, Julia3 on a Jupyter notebook, which you can download here. I wrote up this solution having in mind people like me, who don’t really know anything, but have a generally favourable attitude towards math.

The first step is to brute force search for a rational solution. That’s easy enough, but gives us mostly repetitions of the points $(-1,-1)$, $(1,-1)$, and $(-1,1)$. They make some denominators of the original equation evaluate to zero, and by looking at the graph it is obvious that the point $(-1,-1)$ doesn’t work4, so I excluded them and selected a nice-looking solution as the starting point:
\[P_1 = \left(-\frac5{11},\frac9{11}\right)\]Now, I needed to calculate the slope of the curve at this point. This lies firmly on the straightforward-but-tedious territory5, so I won’t bore you with the details. The result is that
\[ \frac{\mathrm{d}y}{\mathrm{d}x} = \frac{-3x^2+6x+6xy+3y^2+5y+3}{-3x^2-5x-6xy+3y^2-6y-3},\]and with the slope we can easily find the tangent line. Now finding the intersection of this line with the curve is even more straightforward and tedious6, so I’m not going to even type out the answer (for the details check the Wikipedia page linked above). Anyway, now we have the intersection point
\[I = \left(\frac{5165}{9499},-\frac{8784}{9499}\right),\]but we have a problem: if we draw the line between $P_1$ and $I$ we won’t get a new point, as this line intersects the curve only at $P_1$ (twice) and $I$. We could just apply the same procedure that we did to $P_1$: calculate the tangent at $I$ and with that get a new intersection point. This works, but the integers in the numerator and denominator of the numbers we get by iterating this procedure grow hideously fast7, and the computer dies before returning a solution with only positive rationals.

There is an easy way out, though: note that the equation is symmetric under exchange of $x$ and $y$, so we can simply take the new point as
\[P_2 = \left(-\frac{8784}{9499},\frac{5165}{9499}\right),\]and the line through $P_1$ and $P_2$ does intersect the curve at a new point. Taking $P_3$ to be again its flipped version
\[P_3 = \left(-\frac{396650011}{137430135},-\frac{934668779}{137430135}\right),\]we can go on.

A new question does arise. Do we take $P_4$ from the line through $P_2$ and $P_3$, or from the line through $P_1$ and $P_3$? It turns out that both methods work, but the number of digits grows much faster with the former method than with the latter8, so it is unwise to use the former method in general. We iterate away, and find the solution with $P_{13}$, with the resulting integers having more than 150 digits.

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