Increasingly capable Artificial Intelligence (AI) technologies are gaining steam in proving long-standing mathematical conjectures. The latest development involves a proof of another one of prolific mathematician Paul Erdős' 1,135 unsolved problems, but unlike AI's previous accomplishments in tackling part of Erdős' legacy, GPT-5.4's proof of Erdős #1196 appears to be genuinely novel.

The previous proofs by AI systems of other Erdős problems did not answer the question of whether the technology was bringing anything new to the table. The problems themselves could be considered "low-hanging fruit", whose unsolved status had more to do with the obscurity of the conjectures within the Erdős collection than their difficulty. When AI developed proofs for them, it more or less followed the playbook that mathematicians had established in proving other Erdős conjectures.

But Erdős' 1196th conjecture is not in that category. Mathematicians had previously taken on the challenge of developing a proof for it, largely turning to the tools of probability and statistics in the process. But instead of copying that approach, GPT-5.4 Pro found the path for proving the conjecture differently. Mathematician Terrence Tao offered this observation:

I had previously stated the opinion that the AI-generated proof had inadvertently highlighted a tighter connection between the anatomy of integers and the theory of Markov chains than had previously been explicitly noted in the literature. Based on further developments, I would like to update that opinion to the following: the AI-generated proof artefact, when combined with subsequent (and mostly human-generated) analysis, has revealed a tight connection between the anatomy of integers and flow network theory that does not, to my knowledge, have any explicit precursor in the literature (although related uses of Markov chains in adjacent settings do appear in that literature).

The development of the proof was verified using the Lean proof assistant, which we would argue is the secret sauce behind why AI technologies are making such rapid progress in advancing proofs to unsolved problems in the field. The pairing of the technologies is key to the advancement.

Getting back to the novelty of the proof, here's a comment by Jared Duker Lichtman, who developed the first proof of the related Erdős primitive set conjecture as part of his doctoral thesis in 2024.

In my doctorate, I proved the Erdős Primitive Set Conjecture, showing that the primes themselves are maximal among all primitive sets.

This problem will always be in my heart: I worked on it for 4 years (even when my mentors recommended against it!) and loved every minute of it.

[Primitive sets are a vast generalization of the prime numbers: A set S is called primitive if no number in S divides another.]

Now Erdős#1196 is an asymptotic version of Erdős' conjecture, for primitive sets of "large" numbers. It was posed in 1966 by the Hungarian legends Paul Erdős, András Sárközy, and Endre Szemerédi.

I'd been working on it for many years, and consulted/badgered many experts about it, including my mentors Carl Pomerance and James Maynard.

The proof produced by GPT5.4 Pro was quite surprising, since it rejected the "gambit" that was implicit in all works on the subject since Erdős' original 1935 paper. The idea to pass from analysis to probability was so natural & tempting from a human-conceptual point of view, that it obscured a technical possibility to retain (efficient, yet counter-intuitve) analytic terminology throughout, by use of the von Mangoldt function \Lambda(n).

The closest analogy I would give would be that the main openings in chess were well-studied, but AI discovers a new opening line that had been overlooked based on human aesthetics and convention.

In fact, the von Mangoldt function itself is celebrated for it's connection to primes and the Riemann zeta function--but its piecewise definition appears to be odd and unmotivated to students seeing it for the first time. By the same token, in Erdős#1196, the von Mangoldt weights seem odd and unmotivated but turn out to cleverly encode a fundamental identity \sum_{q|n}\Lambda(q) = \log n, which is equivalent to unique factorization of n into primes. This is the exact trick that breaks the analytic issues arising in the "usual opening".

Joshua Zelinsky offers perhaps the best framing of the accomplishment and what it could mean for additional progress:

Four things to note: #1196 is a decently well known problem. It wasn’t like Erdős-Straus level fame, but it is well known enough that I was familiar with it. Second, this is not a problem where no one had worked on it; there was a lot of prior work on it and closely related problems. Third, this is not example where the AI made small modifications to things in the literature or recognized that large parts of the problem were in an obscure paper. The approach the AI used is largely a different direction than the literature on this problem went. Fourth, and closely related to three, this proof does look like parts of it will inspire subsequent proofs because it really is going in a different direction which now looks likely to be a productive line of investigation for similar problems.

If this kind of progress continues, the productivity of AI technology in cracking unsolved math problems will be the biggest math story of the year. AI was already the biggest math story of 2025, but 2026 is shaping up to be even more so.

Once upon a time, and in truth, as recently as four years ago, there was a strong relationship between the spot price of gold and the inflation-indexed market yield of 10-Year Constant Maturity U.S. Treasuries.

It was an inverse relationship. When the inflation-adjusted interest rate on the 10-year bonds fell, signaling an increase in inflation, the price of gold would rise. And vice-versa. If the inflation-adjusted yields of these treasuries rose, indicating falling inflation, the price of gold would fall as well.

That made a sort of sense. But that relationship has broken down in the last four years. Starting from 17 March 2022, when the Federal Reserve finally acted to hike interest rates to combat the high inflation unleashed by the Biden administration a year earlier, the relationship between the inflation-indexed 10-year Treasury and gold spot prices has steadily broken down.

We can see that in the following chart, in which the price of gold has fully decouples from the interest rate of the inflation-protected 10-year Treasury.

Most of this decoupling has taken place since 27 December 2023. At that time, the yield of the inflation-indexed 10-year Treasury was 1.64% and the price of an ounce of gold was $2,079. Since that date, the 10-year inflation-protected Treasury has ranged between that low and a high of 2.28% on 30 April 2024. Today, that yield is just below the middle of that range at 1.88%.

But the price of gold has soared during this time. In recent months, as the 10-year TIPS yield has ranged between, it soared to reach a high of $5,414.49 an ounce on 28 January 2026. In the weeks since, it has plummeted, losing over 10% of its peak value. All without any big change in the inflation-protected treasury yield.

It's like the carnival game "high striker". Gold prices are rising and falling by huge amounts independently of changes in yields and inflation.

In mathematics, the slope of a vertical line is described as "undefined". Which is to say there is no relationship between it and whatever the horizontal axis represents. In the case of gold prices, inflation, and the yield of 10-year U.S. Treasuries, we can say that in March 2026, no such relationship exists between these three things, nor has there been such a relationship in years.

Perhaps there never was. Or perhaps we're missing a bigger factor that's holding greater sway today. If we are, what do you suppose it is and why has it been able to cause the price of gold to change so much in during the last four years as compared to how it changed during the preceding 15 years?

Image Credit: Microsoft Copilot Designer. Prompt: "An editorial cartoon of a Wall Street bull and bear who are taking turns playing the high striker carnival game that is labeled 'GOLD PRICES' and the bear asks 'DIDN'T THIS GAME USE TO HAVE SOMETHING TO DO WITH INFLATION?".

Recent months have seen several announcements in which the Large Language Models (LLM) behind modern Artificial Intelligence (AI) technologies have solved long-standing problems in the field. Among those previously unsolved problems are a handful originally conceived by Paul Erdős, who generated a catalog of 1,135 challenges for mathematicians to take on. Several AI developers target these problems as means of measuring the progress they're making in developing their systems.

But are today's AI systems really capable of solving these unsolved problems? LLMs have been described as an advanced form of an autocomplete function, or even as a "glorified autocorrect" program the AI systems use to statistically predict what response should follow the prompts it has been given based on the reams of data on which it has been trained.

In the case of the Erdős problems that have been solved by AI, there could be something to that argument. The solved problems have been described as being relatively low-hanging fruit, whose unsolved status may have more to with their relative obscurity. Being similar to other Erdős problems that have been solved, which would be part of the training library used by the LLMs, that similarity could be enough to solve them.

The alternative hypothesis is the math-LLMs are genuinely capable of coming up with original solutions for these problems. But how can we tell which hypothesis is closer to the truth?

A group of eleven mathematicians has proposed a interesting experiment to find out. They're tapping their currently unpublished research to remove the possibility that the LLM is effectively rehashing mathematical solution processes to which they have previously been exposed. Here's the abstract for their preprint paper, which they uploaded on 6 February 2026:

To assess the ability of current AI systems to correctly answer research-level mathematics questions, we share a set of ten math questions which have arisen naturally in the research process of the authors. The questions had not been shared publicly until now; the answers are known to the authors of the questions but will remain encrypted for a short time.

"Short time" was one week. The ten questions were unencrypted on 13 February 2026.

That action started a clock for challenging today's math-AI systems to see if they're genuinely capable of autonomously solving mathematical research questions. At this writing, we don't know what results, if any, have been put forward and whether they stands up to scrutiny.

Regardless of how it goes, it's a genuinely exciting research effort for which we're looking forward to learning the outcome.

Image credit: Artificial Intelligence - hand touches icon with symbol of network and brain by Gerd Altmann on PublicDomainPictures.com. Creative Commons Creative Commons - CC0 Public Domain.

Shiraz Robinson is a grad student in data science at the University of Virginia, who came to our attention shortly after the Christmas holiday for his Merry Christmans equations. Unfortunately, that was too late to feature his tradition of blending math formulas and holiday celebrations together, at least for that holiday, but he's back in time for Valentine's Day with another holiday-appropriate math formula!

Here's the equation, suitable for sharing with that special someone who appreciates math as much as you do! And before you ask, yes, it's safe for work....

x² + [y - (x²)¹^/³]² = 1

While the mathematically inclined will immediately interpret this equation of love, others may need to plot it on a graph to truly appreciate its romantic message. We recommend checking out this GeoGebra version because you can play with the parameters d'amour.

Update 18 February 2026: Or, if you prefer to leave the math to others (and use a fancier equation), there's this video!

Previously on Political Calculations

• Picking the Right Date for Valentine's Day
• The Indisputable Inter-Age Rule for Dating
• Happy Valentine's Day!
• The Maths of Love
• The Most Romantic Shape in Mathematics
• A Formula for Love
• Conway's Game of Love

Mathematician Paul Erdős died nearly 30 years ago, but left behind a massive legacy. Famously known for his collaborations with other mathematicians and his eccentric practice of showing up unannounced to their doorsteps and staying only as long as they provided him with interesting challenges to ponder, the prolific mathematician left behind 1,135 problems for succeeding mathematicians to advance the field by either proving or disproving them.

In the last few months, three of those problems have been cracked by Artificial Intelligence (AI) systems harnessed with the well established Lean proof assistant. NeuronDaily tells the story of the third, which is impressive because there was little-to-no online material the AI system could have accessed to develop its proof, which means the AI paired with a proof assistant is generating genuinely original work:

Remember those brain-teaser math problems from school that made you want to throw your pencil? Now imagine ones so hard they've stumped mathematicians for decades. Paul Erdős, who published more papers than anyone in math history, left behind hundreds of these puzzles when he died in 1996. This weekend, GPT-5.2 Pro solved one.

Neel Somani prompted the AI to tackle Erdős Problem #397, which asks whether infinitely many solutions exist for a specific equation involving central binomial coefficients. GPT-5.2 generated the proof, the tool Aristotle formalized it in Lean (a verification language), and Fields Medalist Terence Tao accepted it.

Here's Neel Somani's X tweet announcing the accomplishment:

NeuronDaily explains why the proof of Erdős Problem #397 is a big deal:

It’s part of a wave of autonomous solves: GPT-5.2 has now cracked Problem #728, #729, and 397.

Verified mathematics: The Aristotle system auto-corrected gaps in proofs and produced Lean-verified code.

Self-contained reasoning: Unlike October 2025's GPT-5 controversy (which just found existing literature), Tao says these are original proofs.

While impressive for the technology, the new proofs don't yet represent the kind of ground-breaking accomplishment that would permanently leave human mathematicians in the dust.

The catch? Tao emphasizes these are “lowest hanging fruit”; problems solvable with standard techniques, not profound breakthroughs. GPT-5.2 scores 77% on competition-level math but only 25% on open-ended research requiring genuine insight.

Three years ago, having any computing system score 77% on competition-level math would have been a remarkable achievement, but scoring 25% on open-ended research was a pipe dream. Considering the amazing progress that's been made with the use of AI in maths in just the past three years, what lies in store for the next three years?

Image credit: Paul Erdős and Terrence Tao in 1985. Wikimedia Commons. Creative Commons Attribution-Share Alike 2.0 Generic Deed. We were surprised that Erdős collaborated with Tao when the latter was just 10 years old! Terence Tao, of course, went on to have an equally remarkable career and has an Erdős Number of 2.

The year that was 2025 has all but come and gone, so it's the perfect time to look back at the year's biggest math stories!

For our selection criteria, we've emphasized math stories that involve practical applications in selecting both the contenders and the Biggest Math Story of 2025, which we'll present at the end of this year's edition. That real world connection is something we find essential in exploring achievements in math, which is why we place our main focus on it.

But that doesn't stop us from appreciating the year's "pure" math accomplishments. For example, we're fans of the work by the volunteers behind the Great Internet Mersenne Prime Search project, which is dedicated to identifying these special prime numbers. Last year, they made news when a volunteer found the current record holder for the largest known prime number, M(136,279,841), the 52nd Mersenne prime.

While they didn't find a bigger prime number this year, their work to verify that no other Mersenne primes fit in-between the gaps of their previous discoveries continued. In 2025, they expanded their test results to verify that they had identified all Mersenne primes smaller than M(77,232,917), the 50th Mersenne prime. We anticipate they'll clear the distance between that prime and M(82,589,933), the 51st Mersenne prime, sometime in 2026.

There is however an enormous gap between the 51st Mersenne prime and the current-52nd Mersenne prime that may very well contain one or two additional Mersenne primes. We are very much looking forward to what they find.

That's enough gushing about prime numbers, so let's get to the biggest math stories of the year that was. Starting with....

The Elephant in the Room

2025 was a year in which avoiding any mention of developments related to Artificial Intelligence (AI) models and the Large Language Models that have come to define the machine learning technology was all but impossible. Hundreds of billions of dollars all around the world are being invested to build and develop the infrastructure needed to support advanced AI systems.

Many disciplines felt the impact of AI technology in 2025 and mathematics is one of many in which AI systems are starting to have a profound impact.

One of the more unique stories we encountered is about one of the first valid mathematical proofs ever produced by a generative AI system. That story came to our attention through a post at X and it's probably best to let that post tell the tale:

GPT-5 just casually did new mathematics.

Sebastien Bubeck gave it an open problem from convex optimization, something humans had only partially solved. GPT-5-Pro sat down, reasoned for 17 minutes, and produced a correct proof improving the known bound from 1/L all the way to 1.5/L.

This wasn’t in the paper. It wasn’t online. It wasn’t memorized. It was new math. Verified by Bubeck himself.

Humans later closed the gap at 1.75/L, but GPT-5 independently advanced the frontier. A machine just contributed original research-level mathematics.

If you’re not completely stunned by this, you’re not paying attention.

We’ve officially entered the era where AI isn’t just learning math, it’s creating it.

Here's the X post in which Bubeck announced the accomplishment.

But as Bubeck later notes, while GPT-5 did something that was both new and novel, humans still beat AI to the punch and delivered a better proof focused on the convex optimization problem that outperforms the AI-generated proof:

Now the only reason why I won't post this as an arxiv note, is that the humans actually beat gpt-5 to the punch :-). Namely the arxiv paper has a v2 https://arxiv.org/pdf/2503.10138v2 with an additional author and they closed the gap completely, showing that 1.75/L is the tight bound.

While that better proof clearly beat its accomplishment, the unexpected artificially generated proof demonstrates AI systems are becoming more capable and useful. Because they are, more things are becoming possible.

Later in the year, a paper revealed another major advancement for the use of AI technologies in maths. Here's that story, which involves events that took place in 2024:

At the 2024 International Mathematical Olympiad (IMO), one competitor did so well that it would have been awarded the Silver Prize, except for one thing: it was an AI system. This was the first time AI had achieved a medal-level performance in the competition's history....

The AI is AlphaProof, a sophisticated program developed by Google DeepMind that learns to solve complex mathematical problems. The achievement at the IMO was impressive enough, but what really makes AlphaProof special is its ability to find and correct errors. While large language models (LLMs) can solve math problems, they often can't guarantee the accuracy of their solutions. There may be hidden flaws in their reasoning.

AlphaProof is different because its answers are always 100% correct. That's because it uses a specialized software environment called Lean (originally developed by Microsoft Research) that acts like a strict teacher verifying every logical step. This means the computer itself verifies answers, so its conclusions are trustworthy....

In addition to solving seemingly intractable math problems, AlphaProof could also be employed by mathematicians to correct their work and help them develop new theories.

The integration of a well-established proof assistant in the AI system represents a major advance and directly addresses one of generative AI technology's main shortcomings: these systems can "hallucinate" and produce absolutely garbage results. Kind of like an editorial cartoon featuring an emcee about to announce the winner of The Biggest Math Story of the Year who has too many hands and fingers.

For mathematicians though, that risk has become much smaller because of the strict discipline imposed by the proof assistant. If the attempted results fail to pass muster according to the logic imposed by the proof assistant, they're rejected.

This is probably the last year in which it might be possible to contain AI-related developments within a single section of a year-end wrap up. Fortunately for humans, none of these stories represent the Biggest Math Story of 2025!

Cracking How a Cryptocurrency Collapsed

In 2022, the cryptocurrency Terracoin collapsed. At the time, there were indications it had been the result of a coordinated effort to crash the cryptocurrency by a handful of traders who stood to profit as other investors got wiped out. But the question of exactly how it was done has been a mystery. Three years later, intrepid researchers using advanced mathematical tools and specialized software they developed cracked the mystery of how it was done. Here's an excerpt:

In a new study published in ACM Transactions on the Web, researchers from Queen Mary University of London have unveiled the intricate mechanisms behind one of the most dramatic collapses in the cryptocurrency world: the downfall of the TerraUSD stablecoin and its associated currency, LUNA. Using advanced mathematical techniques and cutting-edge software, the team has identified suspicious trading patterns that suggest a coordinated attack on the ecosystem, leading to a catastrophic loss of $3.5 billion in value virtually overnight.

The study, led by Dr. Richard Clegg and his team, employs temporal multilayer graph analysis—a sophisticated method for examining complex, interconnected systems over time. This approach allowed the researchers to map the relationships between different cryptocurrencies traded on the Ethereum blockchain, revealing how the TerraUSD stablecoin was destabilized by a series of deliberate, large-scale trades.

Stablecoins like TerraUSD are designed to maintain a steady value, typically pegged to a fiat currency like the US dollar. However, in May 2022, TerraUSD and its sister currency, LUNA, experienced a catastrophic collapse. Dr. Clegg's research sheds light on how this happened, uncovering evidence of a coordinated attack by traders who were betting against the system, a practice known as "shorting."

"What we found was extraordinary," says Dr. Clegg. "On the days leading up to the collapse, we observed highly unnatural trading patterns. Instead of the usual spread of transactions across hundreds of traders, we saw a handful of individuals controlling almost the entire market. These patterns are smoking gun evidence of a deliberate attempt to destabilize the system."

From a trading standpoint, the apparent conspiracy to crash the TerraUSD stablecoin was possible because it was very thinly traded and lacked the liquidity needed to offset the determined efforts by those shorting the cryptocurrency. But more importantly, Clegg's team identified and developed the mathematical tools needed to detect such an effort, which will have real world application as digital currencies like stablecoins become more common.

"These kids today, with their loud music and hula hoops!..."

Every year, Ig Nobel Prizes are awarded to unusual scientific research that, given the subjects involved, make people laugh and then make them think. Occasionally, there's a mathematics category. Here's our contender for next year's Ig Nobel Prizes, which features the first ever results that "explain the physics and mathematics of hula hooping":

"We were specifically interested in what kinds of body motions and shapes could successfully hold the hoop up and what physical requirements and restrictions are involved," explains Leif Ristroph, an associate professor at New York University's Courant Institute of Mathematical Sciences and the senior author of the paper, which appears in the Proceedings of the National Academy of Sciences....

The results showed that the exact form of the gyration motion or the cross-section shape of the body (circle versus ellipse) wasn't a factor in hula hooping.

"In all cases, good twirling motions of the hoop around the body could be set up without any special effort," Ristroph explains.

However, keeping a hoop elevated against gravity for a significant period of time was more difficult, requiring a special "body type"—one with a sloping surface as "hips" to provide the proper angle for pushing up the hoop and a curvy form as a "waist" to hold the hoop in place....

The paper's authors conducted mathematical modeling of these dynamics to derive formulas that explained the results—calculations that could be used for other purposes.

"We were surprised that an activity as popular, fun, and healthy as hula hooping wasn't understood even at a basic physics level," says Ristroph.

"As we made progress on the research, we realized that the math and physics involved are very subtle, and the knowledge gained could be useful in inspiring engineering innovations, harvesting energy from vibrations, and improving robotic positioners and movers used in industrial processing and manufacturing."

Makes you laugh, then makes you think. While this hula hooping story may be a perfect contender for the Ig Nobel Prizes, it's not the Biggest Math Story of 2025. For that story, please scroll down to the next section....

Unifying the Math for Describing How Fluids Behave at All Scales of the Physical Universe

In 1900, David Hilbert identified what he thought would be the 23 biggest challenges mathematics had to offer. 125 years later, over half of those challenges are still waiting to be fully resolved but Hilbert's sixth challenge was very likely cracked in 2025.

That challenge lay in defining a single set of mathematical axioms that could unite the different maths needed to describe how fluid particles behave at different scales. At the smallest "microscopic" level, the motion of individual fluid particles can be described by Newton's laws of motion. Scaling up to the medium "mesascopic" level, how groups of fluid particles behave can be described with the Boltzmann equation. Scaling up once again to the largest "macroscopic" level however takes yet another formulation, with Navier-Stokes equations taking over the work of describing how the fluid itself moves.

Since 1900, a good amount progress toward resolving Hilbert's sixth problem has been made by many contributors, but a major gap remained. In March 2025, three mathematicians, Yu Deng, Zaher Hani, and Xiao Ma posted a preprint paper that filled the remaining gap in the efforts over the past 125 years to create a unified set of math to describe fluid motion at all levels. Quanta Magazine covers the achievement in uniting the math describing the quantum-level (microscopic), medium-level (mesascopic) and large (macroscopic)-scale dynamics of how fluid particles interact in the first third of the following 20-minute video that provides an excellent overview of we think is the biggest math story of the year:

We have more background in our earlier coverage of the accomplishment. At this writing, their work is still awaiting verification, but if Deng, Hani and Ma's work holds, successfully connecting the math of Newton, Boltzmann, Navier, and Stokes across these three scales of reality represents a massive achievement in mathematical theory.

If you watch the full video, Hong Wang and Joshua Zahl's proof of the three-dimensional Kakeya Conjecture (or Needle Problem), a challenge dating back more than a century that revolves around how to determine what the smallest volume is in which a needle can be spun around all possible orientations within a space. Wang and Zahl's remarkable accomplishment is a runner up for our criteria of practical application given its links to geometric measure theory and harmonic analysis.

Ultimately, the question of which achievement would claim the title of being the Biggest Math Story of 2025 came down to recognizing Deng, Hani and Ma's proof represents the culmination of a challenge set out 125 years ago to firmly underpin the physics of how fluids behave with math that works at all scales of the physical universe. It's hard to get much bigger than that in 2025.

Previously on Political Calculations

The Biggest Math Story of the Year is how we've traditionally marked the end of our posting year since 2014. Here are links to our previous editions and our coverage of other math stories during 2024:

• The Biggest Math Story of the Year (2014)
• The Biggest Math Story of 2015
• The Biggest Math Story of 2016
• The Biggest Math Story of 2017
• The Biggest Math Story of 2018
• The Biggest Math Story of 2019
• The Biggest Math Story of 2020
• The Biggest Math Story of 2021
• The Biggest Math Story of 2022
• The Biggest Math Story of 2023
• The Biggest Math Story of 2024
• The Biggest Math Story of 2025
• The Next Mersenne Prime
• Conway's Game of Love
• Game Theory and Settling the Debts of the Deceased in Ancient Times
• Cogitata Physico-Mathematica
• Impossible Maths
• What President Trump's "Liberation Day" Tariff Math Really Says
• Breakthrough in Unifying Math to Describe How Fluids Behave at Different Scales
• The Trillion Dollar Equation
• Flipping Burgers on an Infinite Plane
• Outside the Box Thinking: Galileo's Geometry of Dante's Inferno
• A Tetrahedron That Always Lands on the Same Side
• AI Delivers an Unexpected Mathematical Proof
• The Geometry Problem Tha6t Could Have Protected the French Royal Jewels
• Gift Ideas for the Math Lovers You Know

This is Political Calculations' final post for 2025. Thank you visiting, we hope found the stories and analysis we've presented throughout the year to be either thought-provoking, informative, or entertaining. We'll see you again in the New Year, which we'll kick off with our annual tradition of presenting a tool to help you find out what your paycheck will look like in 2026.

Image credit: Microsoft Copilot Designer. Prompt: "An editorial cartoon featuring an emcee opening an envelope to announce the biggest math story of the year".

It can be very difficult to find a good gift to give the people closest to you. That challenge can become even more difficult if they're the kind of people who really like math. What can you give to them that will excite them and, perhaps most importantly for the most enthusiastic among them, they haven't already gotten for themselves? Or that you haven't gotten them before?

Speaking of which, let's recap the very short list of gift ideas we've previously suggested, in case you've already exhausted those options:

• What to Get a Maths Enthusiast for Christmas (Klein bottle)
• Oddly shaped objects that roll smoothly (oloids)
• Games Mathematicians Play (21X)

This year, we'll suggest three new options, including two that offer the bonus of being potentially practical. Starting with....

The "Sometimes I Go Off on a Tangent" Coffee Mug

Coffee mugs are inherently practical, and frankly, much easier to drink from than a Klein bottle. This one features a design than illustrates the mathematical concept of tangents, pairing it with a really bad math pun.

Despite the bad mathematical pun, this 11-ounce ceramic mug is microwave and "dishwasher safe", though handwashing is recommended. Otherwise, it's a fully practical mug.

Many Amazon reviewers mention buying this mug as a gift for their math teachers. However, if you really knew any math teachers, you would know that 11-ounces of a caffeine-laden beverage isn't going to cut it for them. Instead, this is a mug in which they can keep their fancy Hagoromo colored chalk on their classroom desk. Right next to the much larger mug from which they will consume their preferred caffeinated beverages.

"The Proof Is in the Pudding" Bowls

These are actual pudding bowls which were created and marketed by "The Unemployed Philosophers Guild", who love both bad puns and mathematical proofs. Here's an excerpt of their product description:

• A set of four ceramic pudding bowls with the proofs to classic theorems of Euclid, Hippasus, Pythagoras, and Gauss. Terrific gift for mathematicians, scientists, students, or any geek on your gift list.
• We are not going to lie to you: this one is for the mathematikoi-the inner circle of the school of Pythagoras. If you think all math is rational, you are living in a fool's paradise.
• These bowls are dangerous-so dangerous that one theorem just might have culminated in murder. (Look it up-Hippasus of Metapontum sleeps with the fishes because he couldn't keep his mouth shut about the square root of 2).

They're not kidding when they say these bowls are for the "mathematikoi", which we think of as those people who like math so much they unironically get tattoos of their favorite formulas. If you know that person, you now know what to get them as a gift.

Infinite Powers: How Calculus Reveals the Secrets of the Universe

This is a good introductory book on the topic of calculus, why and how it was invented, and what people use it for. That said, the best and most engaging parts of the book are the ones where author Steven Strogatz breaks away from the math and has fun with the topic at hand. Here's an example:

All across the world, students are being taught that division by zero is forbidden. They should feel shocked that such a taboo exists. Numbers are supposed to be orderly and well behaved. Math class is a place for logic and reasoning. And yet it's possible to ask simple things of numbers that just don't work or make sense. Dividing by zero is one of them.

The root of the problem is infinity. Dividing by zero summons infinity in much the same way that a Ouija board supposedly summons spirits from another realm. It's risky. Don't go there.

Strogatz is also the host of Quanta Magazine's "The Joy of Why" podcast, the title of which is a riff on his earlier book "The Joy of X: A Guided Tour of Math, from One to Infinity. Infinite Powers is very rare math book that falls in between the binary extremes of "too simple" and "too advanced" levels of knowledge needed to enjoy reading them. As such, this book represents a better-than-average gift idea for the majority of maths enthusiasts you may know.

With these gift ideas in hand for perhaps the most difficult person for whom you need to give a present, you can now move on to shop for more normal people!

Image credit: Hands exchanging a gift with a red ribbon photo by Vitaly Gariev on Unsplash.

Imagine you're in charge of placing security cameras inside a gallery filled with valuable objects. Your boss is willing to pay to put in as many cameras at it will take to protect the gallery's contents, provided you meet two conditions:

1. The cameras must collectively provide 100% visual coverage of the entire area of the gallery.
2. You have to determine the minimum number of cameras that will be needed to do the job.

Believe it or not, this is a famous 50-year old geometry problem that's known as the art gallery problem. In the following 50-minute video, CC Academy describes how the problem can be solved using graph theory for any strangely shaped floorplan:

Kit Yates discusses why the recent heist of the French royal jewels from the Louvre Museum in Paris brings home the importance of this geometry problem:

At a hearing in front of the French Senate in the immediate aftermath of the robbery, Laurence des Cars, the director of the world famous museum, admitted that the museum had "failed to protect" the crown jewels. She admitted that the only camera covering the balcony the thieves used was facing the wrong way and a preliminary report revealed one in three rooms in the Denon wing where the thieves struck had no security cameras. More generally Des Cars acknowledged that cuts in surveillance and security staff had left the museum vulnerable and insisted that the Louvre's security system must be reinforced to "look everywhere".

Alarms at the museum apparently sounded as they should, according to the French culture ministry. Yet it is the third high profile theft from French museums in two months, which have left the ministry implementing new security plans across France.

As they should, if for no other reason than because their security scheme has been shown to have gaping holes that thieves an easily drive a Böcker Agilo truck-mounted moving lift through.

But more to the point, had those responsible for overseeing the Louvre's security solved their specific version of the art gallery problem, the thieves' plans might have been thwarted. Yates concisely explains how to quickly determine how many cameras you might need using only geometric principles:

The answer, it turns out, depends on the number of corners (or, as mathematicians call them, "vertices"), as there will be as many walls as there are corners in a room. Some simple division helps us work out how many cameras are needed.

By dividing the number of corners in a room by three, that will tell us how many cameras are needed to cover it, assuming they have a full 360 degree field of view....

This even works if the number of corners isn't neatly divisible by three. For a 20-sided gallery, for example, the answer works out at six and two thirds. In these cases you can take the whole number – so we'd never need more than six cameras in a 20-sided room.

Yates continues by getting into the graph theory that tells where those 360-degree view cameras can then be optimally placed.

In 1978 Steve Fisk, a mathematics professor at Bowdoin college in Maine, US, came up with a proof – considered one of the most elegant in all of mathematics – of this lower limit on the number of cameras needed.

His strategy was to divide the gallery up into triangles (check out the left image of the figure below). He then proved that you can pick just three colours – say red, yellow and blue – and assign a different colour to the corners of each triangle. This would mean that every triangle in your gallery has a different colour in its three corners (See the right image of the figure below for an example). This is known as "three-colouring" the corners.

Triangles are one of those "convex" polygons we mentioned earlier, so a camera positioned at any corner (or indeed anywhere in the triangle) can see every point in that shape. Every triangle has corners with each of the three colours. That means you can pick just one of the colours and place cameras at those positions. Those cameras will be able to see every part of every triangle, and hence every part of the gallery. But here's the best part.

The beauty of Fisk's proof is you can just choose the colour with the fewest dots, and you'll still cover the whole gallery. In the 15-sided shape above, by choosing the red dots, we can get away with only four cameras.

What are the chances that all of France's art gallery security problems are now homework assignments for the country's undergraduate math students taking courses in either geometry or graph theory?

Image credit: Art Gallery Problem sample with 4 cameras by Rocchini on Wikimedia Commons Creative Commons CC BY 3.0 Attribution 3.0 Unported Deed.

For all the press that AI and all the companies behind it get, it's rare to find any story that identifies something the Large Language Models (LLMs) behind today's Artificial Intelligence technologies have accomplished that represents a true advancement.

Most stories are about AI's teething problems, such as its problems in creating images of people with too many figures or some other form of body horror dystopia. Other news items deal with how the ability of AI to completely automate writing is negatively impacting education, publishing, music, filmmaking, and other fields.

But stories involving AI systems doing anything new and useful, that hasn't been seen or done before, are rare.

That story still hasn't been written. At least in any traditional media. But there is breaking news of an AI system that's broken new ground in mathematics. By generating a new mathematical proof that mathematicians have verified is correct

That story came to our attention through a post at X and it's probably best to let that post tell the tale:

GPT-5 just casually did new mathematics.

Sebastien Bubeck gave it an open problem from convex optimization, something humans had only partially solved. GPT-5-Pro sat down, reasoned for 17 minutes, and produced a correct proof improving the known bound from 1/L all the way to 1.5/L.

This wasn’t in the paper. It wasn’t online. It wasn’t memorized. It was new math. Verified by Bubeck himself.

Humans later closed the gap at 1.75/L, but GPT-5 independently advanced the frontier. A machine just contributed original research-level mathematics.

If you’re not completely stunned by this, you’re not paying attention.

We’ve officially entered the era where AI isn’t just learning math, it’s creating it.

Here's the X post in which Bubeck announced the accomplishment.

But as Bubeck later notes, while GPT-5 did something that was both new and novel, humans still beat AI to the punch and delivered a better proof focused on the convex optimization problem that outperforms the AI-generated proof:

Now the only reason why I won't post this as an arxiv note, is that the humans actually beat gpt-5 to the punch :-). Namely the arxiv paper has a v2 https://arxiv.org/pdf/2503.10138v2 with an additional author and they closed the gap completely, showing that 1.75/L is the tight bound.

While that better proof clearly beat it, the unexpected artificially generated proof demonstrates AI systems are becoming more capable and useful. Because they are, more things are becoming possible.

The open question however is when will AI's promise and ability to deliver on it outshine all the AI-generated slop that dominates what it has done to date?

Image Credit: Microsoft Copilot Designer. Prompt: "An image illustrating the concept of an artificial intelligence system creating an entirely new mathematical proof".

How many times has the following scenario happened to you?

You playing with a tetrahedron, one of those strange geometrical objects whose defining characteristic is that it has four flat sides that are all shaped like triangles, when you suddenly drop it. As it falls, you starting placing mental bets on which of the four sides will land on. Will the side you pick win?

If you're like most people, the answer to the question of how many times this scenario has happeend to you is almost never. Not only do you not run into random tetrahedrons to play with, the game of dropping one to see which side will be on top is not a very fun one to play. The only way to make it interesting is to gamble on the outcome, which makes it a bad dice game, but one with better odds. Instead of a one in six chance of getting it right, you have a one in four chance.

But if you're going to put money down to play this incredibly boring game with the equivalent of a four-sided die, would you really want to lose on average three times out of every four drops?

Of course not! If you want to win all the time, you're going to have to figure out a way to transform your tetrahedron into a loaded die, one that always comes up the way you want it to. Sure, that's cheating, but if money is on the line in playing this game, you want to be the one collecting it from the rubes you might be playing with. How can you rig the game so your tetrahedron always comes out with your winning side on top?

Good news, everyone! Mathematicians have figured out how you can always beat this game. Stand Up Maths' Matt Parker breaks the news in the following video...

And there you have it, a loaded tetrahedron that you can use like a loaded die to always win bets related to which side will it land on if you drop it!

For more information about how this achievement was realized, check out Quanta Magazine's article on the discovery of the monostable tetrahedron!

Long before Galileo Galilei ever thought of pointing the new-fangled invention of the telescope to the night sky, he had a big problem. He needed to get a job.

The year was 1588 and the 24-year-old was seeking to establish himself as a mathematician. Early in the year, he had applied to be the chair of mathematics at the University of Bologna but was rejected. He then set his sights on obtaining an appointment to the University of Pisa, but had to impress an influential committee who could make or break the future for his proposed career.

That group was a literary society. The members of the Florentine Academy had the ability to influence the University of Pisa's hiring of academics and they asked Galileo to deliver two lectures on the geometry of the underworld described in Dante Alighieri's epic poem, The Divine Comedy.

Between these lectures and the publication of a manuscript with theorems he had developed on centers of gravity, Galileo impressed the people he needed and secured a position as chair of mathematics at the University of Pisa. In Gresham College's 2025 Provost Lecture, mathematics professor Sarah Hart explored how several of Galileo's deductions derived from his mathematical analysis of Dante's literary work influenced his later research. The following video presents several remarkable examples of Galileo's creative outside-the-box-thinking that resulted from his taking on the challenge:

There are many lessons here about the value of boundary crossing interdisciplinary research. Hart's lecture also provides a great overview to how Galileo's creative endeavor affected later work in the arts and sciences. Like the examples of fractal patterns she includes near the end of the lecture, there's a seemingly infinite number of rabbit holes to go down to see where they lead.

Summer is a season for hamburgers. But when it's hot and humid out, how much time do you want to spend outside grilling hamburgers? How fast can you fully cook a hamburger patty?

That seems like a simple question, but if you want to use math and the physics of heat transfer to work out the fastest way to cook your hamburgers, it presents a lot of variables you have to take into account. Unless, that is, you simplify the problem in ways that can eliminate them.

For instance, how fast a hamburger cooks will be influenced by the shape of the patty to name just one factor. You could go for a traditional disc-shaped patty or even a Wendy's-style square-shaped patty. But how those burgers cook will be influenced by how heat from the grill 'wraps' around them even as it cooks through the meat from below. It's not hard to have the edges of a burger be the most well-done portion of it, but that may mean the inside of the burger isn't as well-cooked as you might like. How can you ensure your burger is evenly cooked?

For a mathematician or physicist, one way to remove the patty shape as a variable is to assume the burger will be grilled on an infinite plane that only allows the meat to be exposed to heat from the grill on one side at a time. With that assumption, you can count on having the burger be cooked evenly through at all parts. When you do that, you also simplify the problem to be one in which the only variable you have to consider is how often you will flip the burger while cooking it.

That's what one mathematician did in a paper published earlier this year. Here's a plain English summary of what they found when they did the math:

Craving a hamburger but in a hurry? If you wish to find out the quickest way to cook a patty, mathematics can actually help you.

Jean-Luc Thiffeault, a professor of applied mathematics at the University of Wisconsin in Madison, has found that the timing between flips and the number of flips hold the key to the fastest way to cook a meat patty. Moreover, increasing the frequency of flips reduces cooking time by nearly a third.

In his study, published in Physica D on June 17, the researcher used mathematics to study how heat moves through a slab of meat, which simultaneously cooks on the bottom side and cools on the top until the meat is flipped. Eventually, his analysis showed that flipping the patty heats the meat evenly, therefore speeding up the cooking process. He also found that more flips lead to faster cooking.

According to the study, a theoretical 1-centimeter-thick patty that's flipped just once is cooked in 80 seconds. On the other hand, flipping it 10 times at intervals ranging from six to 11 seconds results in a cooking time of only 69 seconds. A maximum decrease of 29% in cooking time is then observed when the patties are continuously flipped.

However, after a threshold is reached, the effect of the number of flips on cooking time becomes inconsequential.

"After three or four flips, the gain in time is negligible," Thiffeault said.

But that's for flipping a burger that fills a grill whose surface is an infinite plane. How does that compare with real-world burger flipping?

As it happens, someone has flipped enough real-world burgers to verify the math-only results:

The study's findings resonate with the observations chef and food writer J. Kenji López-Alt shared in a 2019 article. In the said piece, López-Alt compared the time it took for a burger's internal temperature to reach about 52° Celsius using a certain cooking method he followed. He then realized that flipping the patty every 15 seconds, compared to just once, shortened cooking time by nearly a third.

So there you have it. Flipping a patty once every 15 seconds will get your burgers cooked through about as fast as you can hope to cook it. How many flips you need will ultimately come down to how well-done you want your burgers to be cooked.

Image Credit: A person is grilling a hamburger on a grill photo by Amirhossein Shirdelan on Unsplash.

Ever since the call option was invented by Thales of Miletus about 26 centuries ago, savvy investors have used these and similar financial instruments to make money.

But even though options have been around for a very long time, it wasn't until the last 12 decades and mostly within the last 60 years, in which the math needed to determine what their price should be was finally developed.

In the following 31-minute video, Veritasium's Derek Muller explains the origins of options and the Nobel-prize winning development of the math behind what became the world's first trillion dollar equation. Which turned out to be closely related to the math physicists use to describe the diffusion of heat.

This being a video from the modern internet, there's a mattress commercial built into the middle of the video. If you jump to the 16:02 mark after it starts, it is something you can skip past, unless you're perhaps in the market for a mattress.

In any case, the story is fascinating, because it also involves the one of the most successful investment managers of all time, a mathematician who earned that title by betting against the widely believed efficient market hypothesis by uncovering not-so-random patterns within it and using options to realize gains to beat the market year after year.

For 125 years, mathematicians and physicists have chased after a single solution to a very difficult problem in fluid mechanics.

That problem arises because the math needed to accurately describe the flow of a fluid depends very much on the scale of the physical world being modeled. There's one set of fluid math equations that works at the microscopic level, where individual atoms and molecules of fluids interact. There's another set at the macroscopic level, which involves the combined interactions of millions and billions of fluid particles. And there's a third set at the mesoscopic level, which falls in between the two other scales and demands its own set of equations to describe.

The following video by brain truffle is one of the better introductions we found to the challenges of using math to describe how fluids behave at each of these scales. If you want to skip over some of the foundational discussion, jump ahead to the 16:48 mark.

The challenge lies mathematically unifying the physical theories behind the fluid mechanics at the microscopic, mesoscopic, and macroscopic levels. It was originally proposed in 1900 by mathematician David Hilbert as the sixth of twenty-three problems he identified as worthwhile endeavors in which to pursue mathematically rigorous solutions.

The benefits of establishing such a rigorous solution would mean much greater confidence in the computational analysis supporting how fluids behave in the real world, no matter the scale of the application.

In the twelve full decades since, progress toward establishing that rigorous math to the different scales of fluid mechanics has come in several different waves. The latest wave however could represent the biggest math story of the year because it potentially achieves a substantial portion of Hilbert's goal for this particular problem.

In March mathematicians Yu Deng of the University of Chicago and Zaher Hani and Xiao Ma of the University of Michigan posted a new paper to the preprint server arXiv.org that claims to have cracked one of these goals. If their work withstands scrutiny, it will mark a major stride toward grounding physics in math and may open the door to analogous breakthroughs in other areas of physics....

The new proof broadly consists of three steps: derive the macroscopic theory from the mesoscopic one; derive the mesoscopic theory from the microscopic one; and then stitch them together in a single derivation of the macroscopic laws all the way from the microscopic ones.

The first step was previously understood, and even Hilbert himself contributed to it. Deriving the mesoscopic from the microscopic, on the other hand, has been much more mathematically challenging. Remember, the mesoscopic setting is about the collective behavior of vast numbers of particles. So Deng, Hani and Ma looked at what happens to Newton’s equations as the number of individual particles colliding and ricocheting grows to infinity and their size shrinks to zero. They proved that when you stretch Newton’s equations to these extremes, the statistical behavior of the system—or the likely behavior of a “typical” particle in the fluid—converges to the solution of the Boltzmann equation. This step forms a bridge by deriving the mesoscopic math from the extremal behavior of the microscopic math.

The major hurdle in this step concerned the length of time that the equations were modeling. It was already known how to derive the Boltzmann equation from Newton’s laws on very short timescales, but that doesn’t suffice for Hilbert’s program, because real-world fluids can flow for any stretch of time. With longer timescales comes more complexity: more collisions take place, and the whole history of a particle’s interactions might bear on its current behavior. The authors overcame this by doing careful accounting of just how much a particle’s history affects its present and leveraging new mathematical techniques to argue that the cumulative effects of prior collisions remain small.

Gluing together their long-timescale breakthrough with previous work on deriving the Euler and Navier-Stokes equations from the Boltzmann equation unifies three theories of fluid dynamics.

That's the practical upshot of Deng, Hani and Ma's work, which is now being put to peer review. If you want to know more about the technical aspects of what they did, they spoke about their work in the following video from a Mathematics Colloquium at the University of Chicago in April 2025:

Finally, if you'd like more background into the governing equations of fluid dynamics at the mesoscopic and macroscopic scales, we'll recommend Mojtaba Maali Amiri's 42-minute discussion of them:

If Deng, Hani and Ma's work holds, successfully connecting the math of Newton, Boltzmann, Navier, and Stokes across these three scales of reality represents a massive achievement in mathematical theory, one with practical benefit.

Reference

Yu Deng, Zaher Hani, and Xiao Ma. Hilbert's Sixth Problem: Derivation of Fluid Equations via Boltzmann's Kinetic Theory. Preprint: arXiv:2053.01800v1. [PDF Document]. 3 March 2025.