Top 10 Math Discoveries of 2025

The 10 biggest math breakthroughs of 2025

by Clara Moskowitz Edited by Andrea Thompson

This year has seen some amazing progress in fundamental mathematics. Researchers have made breakthroughs in geometry, topology, chaos theory and more. And three surprising discoveries in our top 10 discoveries involve perennially fascinating prime numbers.

Without further ado, here are some of the most fascinating math findings scientific American Wrote about 2025:

On supporting science journalism

If you enjoyed this article, consider supporting our award-winning journalism Subscribing By purchasing a subscription, you are helping ensure a future of impactful stories about the discoveries and ideas shaping our world today.

a new shape

A new shape called nopterhedron has 90 vertices, 240 edges and 152 faces. The Baroque shape has a surprising property that refutes a long-standing geometric conjecture: no matter how you move or rotate it, a neoparthedron cannot fall through a straight hole into a similar neoparthedron.

prime number pattern

Prime numbers, divisible only by themselves and 1, have long fascinated mathematicians. As you reach larger numbers, it becomes harder to make new discoveries. But this year mathematicians have discovered a set of probabilistic patterns that govern how prime numbers are distributed. Patterns include random chaotic behavior and fractals.

a grand unified theory

A “serious” effort involving nine mathematicians and five papers spanning nearly 1,000 pages has recently proved the geometric Langlands Conjecture. The conjecture combines the properties of various Riemann surfaces, which are coordinate structures that have real and imaginary parts. It is part of a broader set of problems called the Langlands Program, which, if completely proven, could provide a “grand unified theory of mathematics”.

Complication of lump

A long-standing conjecture states that if you join the ends of two different knots together, the complexity of the new knot you create will be the sum of the complexity of the individual knots. But the recent discovery of a knot that is simpler than the sum of its parts disproves that notion.

Fibonacci problems

The Fibonacci sequence, in which each term is the sum of the previous two (1, 1, 2, 3, 5, 8, 13, …), appears throughout nature. And now mathematicians have found that it also provides the answer to a variation of a classic question called the pick-up stick problem: If you have a number of sticks with random lengths between 0 and 1, what is the probability that no three of those sticks can form a triangle?

finding primes

Largest known prime number, 2^136,279,841 − 1 is 41,024,320 digits long, but mathematicians are not satisfied – they want to find even larger primes. This year a team identified a new approach to finding unknown primes. Strategy involves division, or ways in which numbers can be combined to form other numbers.

125 year old problem solved

In 1900 the mathematician David Hilbert presented a series of major unsolved problems. The goal of one of them was to determine the minimum possible mathematical assumptions behind the laws of physics. Researchers later broke the task into sub-goals, and this year the mathematicians claimed to have accomplished one of them: They unified three physical theories to explain the motion of fluids. If the achievement is confirmed, it would be a major step toward solving Hilbert’s sixth problem.

triangle to square

How many pieces do you need to rearrange a triangle into a square? In 1902 a newspaper reader discovered a way to do it in four pieces, but since then no one has managed to do it in fewer pieces. This year researchers finally proved that a triangle cut into fewer than four pieces cannot be turned into a square.

sofa on the go

Anyone who has moved house can appreciate the dilemma of fitting a large sofa in a corner. Mathematicians formally identified this question nearly 60 years ago when they dubbed it the “moving sofa problem”: What is the largest shape that can rotate at right angles down a narrow corridor without getting stuck? Researchers have now found a solution.

catching prime numbers

Another breakthrough on the prime front is a new method of estimating how many prime numbers exist within any given range of numbers. The strategy depends on first eliminating all numbers that are multiples of other primes and therefore cannot be primes themselves. It then counts numbers that are crossed off the list more than once. The study authors also discovered a limit to how accurate any such estimate could be, suggesting that the fundamental mysteries of prime numbers may remain elusive, at least for now.

It’s time to stand up for science

If you enjoyed this article, I would like to ask for your support. scientific American He has served as an advocate for science and industry for 180 years, and right now may be the most important moment in that two-century history.

i have been one scientific American I’ve been a member since I was 12, and it’s helped shape the way I see the world. Science Always educates and delights me, and inspires a sense of awe for our vast, beautiful universe. I hope it does the same for you.

if you agree scientific AmericanYou help ensure that our coverage focuses on meaningful research and discovery; We have the resources to report on decisions that put laboratories across America at risk; And that we support both emerging and working scientists at a time when the value of science is too often recognised.

In return, you get the news you need, Captivating podcasts, great infographics, Don’t miss the newsletter, be sure to watch the video, Challenging games, and the best writing and reporting from the world of science. you can even Gift a membership to someone.

There has never been a more important time for us to stand up and show why science matters. I hope you will support us in that mission.