Recenty, I watched Talking about his calculation, a colleague particle physicist said that it was a calculated he had advanced to an unprecedented level of precision. What was his tool? A 1980s-era computer program called FORM.

The longest equations used by particle physicists are some of the most complex in science. They use thousands upon thousands of Feynman Diagrams to show possible collision outcomes at Large Hadron Collider. Combining formulas such as these is not possible with pen and paper. Even adding them to computers can be difficult. While the school algebra rules are efficient enough to do homework, their efficiency for particle physics is woefully inadequate.

Computer algebra systems are programs that attempt to solve these problems. For 33 years, FORM has been the best program to help you solve the most difficult equations.

Jos Vermaseren is a Dutch particle physicist who developed FORM. This key component of the infrastructure of particle physics allows for complex calculations. However, as with surprisingly many essential pieces of digital infrastructure, FORM’s maintenance rests largely on one person: Vermaseren himself. He has started to withdraw from the FORM development process at age 73. Because academia rewards publications, and not software tools, there has been no replacement. The situation may not improve, and particle physics could be subject to a dramatic slowdown.

FORM began in the 1980s as computers changed rapidly. Schoonschip by Martinus Vermaseren was the predecessor of FORM. This program, which Martinus Veltman created, is a chip designed to be plugged in to an Atari. Vermaseren wanted to create a program that was more readily available and accessible by all universities. It was written in FORTRAN (for Formula Translation) and he began programming it. FORM is a riff. Vermaseren published his software in 1989. By the early ’90s, over 200 institutions around the world had downloaded it, and the number kept climbing.

Every few days since 2000, a paper in particle physics that cites FORM was published on an average basis. “Most of the [high-precision] results that our group obtained in the past 20 years were heavily based on FORM code,” said Thomas Gehrmann, a professor at the University of Zurich.

Some of FORM’s popularity came from specialized algorithms that were built up over the years, such as a trick for quickly multiplying certain pieces of a Feynman diagram, and a procedure for rearranging equations to have as few multiplications and additions as possible. But FORM’s oldest and most powerful advantage is how it handles memory.

As humans can have short-term memory and long-term memory, so computers also have main and external memories. Main memory—your computer’s RAM—is easy to access on the fly but limited in size. While external memory devices, such as solid-state drives or hard disks hold much more information they are also slower. You need main memory to be able to solve long equations.

In the ’80s, both types of memory were limited. “FORM was built in a time when there was almost no memory, and also no disk space—basically there was nothing,” said Ben Ruijl, a former student of Vermaseren’s and FORM developer who is now a postdoctoral researcher at the Swiss Federal Institute of Technology Zurich. It was a problem because the equations required too much memory. For one to be calculated, the operating system would treat the hard disk like main memory. The operating system, not knowing how big to expect your equation to be, would store the data in a collection of “pages” on the hard disk, frequently switching between them as different pieces were needed—an inefficient process called swapping.

FORM does not use swapping, and instead uses its own approach. The program allocates a specific amount of hard drive space to each term when you use FORM for an equation. The software can keep better track of the parts of the equation by using this technique. This makes it possible to quickly bring the pieces to main memory, without having to access any other data.

Memory has grown since FORM’s early days, from 128 kilobytes of RAM in the Atari 130XE in 1985 to 128 gigabytes of RAM in my souped-up desktop—a millionfold improvement. Vermaseren’s tricks are important. Particle physicists are increasingly dependent on precision as they comb through the Large Hadron Collider’s petabytes to find evidence for new particles. This increases the complexity of their equations.

For centuries, mathematicians Fluid motion has been studied and modeled by researchers. Scientists have been able to use equations to explain how water ripples create a pond’s surface. This has allowed them better predict weather patterns, make better planes and understand how blood flows through the circulation system. If written in the proper mathematical language, they can appear very simple. The solutions to these equations are complex enough that it can be difficult for even the most basic questions to be understood.

Leonhard Euler’s equations are the most well-known and oldest of all these equations. They describe the flow and pressure of an ideal incompressible fluid. A fluid that has no viscosity and internal friction is impossible to compress into smaller volumes. “Almost all nonlinear fluid equations are kind of derived from the Euler equations,” said Tarek Elgindi, a mathematician at Duke University. “They’re the first ones, you could say.”

Yet much remains unknown about the Euler equations—including whether they’re always an accurate model of ideal fluid flow. One of the central problems in fluid dynamics is to figure out if the equations ever fail, outputting nonsensical values that render them unable to predict a fluid’s future states.

For decades mathematicians believed that equations could be broken down if they had the right initial conditions. But they haven’t been able to prove it.

Two mathematicians from the University of Michigan have demonstrated that certain Euler equations fail sometimes in an online preprint published October. The proof marks a major breakthrough—and while it doesn’t completely solve the problem for the more general version of the equations, it offers hope that such a solution is finally within reach. “It’s an amazing result,” said Tristan Buckmaster, a mathematician at the University of Maryland who was not involved in the work. “There are no results of its kind in the literature.”

There’s just one catch.

The 177-page proof—the result of a decade-long research program—makes significant use of computers. It is therefore difficult to prove it by other mathematicians. Although they’re still trying to verify it, experts are confident that the work will prove correct. It also forces them to reckon with philosophical questions about what a “proof” is, and what it will mean if the only viable way to solve such important questions going forward is with the help of computers.

Looking for the Beast

The Euler equations can be used to forecast how fluids will change over time if we know where and at what velocity each particle is located in fluid. But mathematicians want to know if that’s actually the case. It is possible that in certain situations the equations can produce exact fluid state values at every moment. But one of these values could suddenly rise to infinity. At that point, the Euler equations are said to give rise to a “singularity”—or, more dramatically, to “blow up.”

Once they hit that singularity, the equations will no longer be able to compute the fluid’s flow. But “as of a few years ago, what people were able to do fell very, very far short of [proving blowup],” said Charlie Fefferman, a mathematician at Princeton University.

It gets even more complicated if you’re trying to model a fluid that has viscosity (as almost all real-world fluids do). The Clay Mathematics Institute is offering a million-dollar Millennium Prize to anyone who proves similar failures in the Navier–Stokes equations. This generalization of Euler equations accounts for viscosity.