At 17, Hannah Cairo solved a major math mystery
quantamagazine.org
Related: https://news.ycombinator.com/item?id=44481441 - July 2025 (105 comments)
quantamagazine.org
“There was this inescapable sameness, in a way. No matter what I did, I was in the same place doing mostly the same things,” she said. “I was very isolated, and nothing I could do could really change that. I’d wake up on certain days and realize, I’m just older.”
I can see the point of sameness in homeschooling, but compared to traditional education? I’m not sure how much flexibility one would have to teach oneself calculus by 11 or the equivalent of an undergrad in math by 14!
That flexibility must be found in something non-traditional!
I’m no prodigy at all whatsoever but school was mostly dull and filled with teenager drama! Nobody knew what Linux was, cared about music production or anything interesting! The talk was which boy/girl whatever
I am learning bit and pieces on how to do this but it's a long and uncertain process. Conflict monitoring is willpower is a new concept to me but it gave me a framework on how to improve my discipline over time.
And with social media, you aren't just comparing yourself to your classmates, but with basically the entire world
1) Forums like askanexper(can't remember the exact name) 2) great books like sams teach yourself in 24 hours and a habit to spend time in library 3) most importantly not as much competition as we have today. There were like a 2-3 kids in my schools of 2k who were into programming. Today that would be approaching half.
>A few more months in prison and I would have proved all the great theorems of the time.
André Weil
This is both unsurprising and shocking to me at the same time.
For institutions of allegedly pure higher learning in a field where it's known that youth is where the advancement happens, the fact 80% axle wrap over a piece of paper that, let's face it, in modern times of grade inflation is pretty much worthless of anything beyond money and sitting in a seat for four years.
Even as a student, I'd be more interested in which professors at Johns Hopkins were accepting students, than which school.
Also, I've seen a great deal of adult babies in academia, so let's not be ageist here.
but she'd be the same age as all the first years on campus, any university must be institutionally equipped to deal with a 17/18 year old. seems like an odd choice to personally intervene to cancel an admission.
like presumably if she enrolled in a bachelor's program, and then signed up for grad math classes while doing independent research, this would be fine. but if she takes the same classes and does the same research for PhD credits instead, no good?
I've personally seen universities go both ways, and it comes down both to individuals, and to the culture of the department/university faculty and administration.
(Not to the culture of the student body, which is influenced by the administration culture, but has very little institutional memory, and almost zero power. If you draw an analogy to nations, there might be one with the most awful 'leaders' seizing and abusing power, but that's "way above the pay grade" of the many nice citizens you will meet -- who didn't know what they were being born into, and will do their best to be decent to each other, despite whatever bits they're unfortunate to learn about the upper powers.)
No professor or university Dean can keep her from doing well on both (1) and (2). It looks like (1) would be easy enough for her. The work she has already done may satisfy (2). Once she has done well on (1) and (2), tough not to award her a Ph.D.
- home schooled? check
This on top of her extraordinary talent and hard work. Institutional education truly is a great leveler, at both the top and bottom.
“Cairo grew up in Nassau, the Bahamas, where her parents had moved so that her dad could take a job as a software developer”
“Travel restrictions stranded her family at her grandparents’ house in Chicago. While they were there, she joined the Math Circles of Chicago”
This doesn’t read like an immigrant. It kind of reads like her dad is a fully American finance dev.
fascinating that the family follows the kids’ educational steps.
Wait that would not even prove anything.
Are there no mathematicians who solved major math mysteries that went to school?
https://readlion.com/homeschoolers-dominate-inland-northwest...
It appears that homeschooling enables students to pursue singular passions further than students in schools. It’s not for everyone though.
My point was just to note some other interesting characteristics beyond the age and gender that show up in coverage.
I wish her the best in her coming career.
A 17-year-old teen refutes a mathematical conjecture proposed 40 years ago - https://news.ycombinator.com/item?id=44481441 - July 2025 (105 comments)
From the article:
https://www.quantamagazine.org/wp-content/uploads/2025/08/Ha...
> I said, "What?"
> Slater said, "Why do you think you should go to graduate school at MIT?"
> "Because MIT is the best school for science in the country."
> "You think that?"
> "Yeah."
> "That's why you should go to some other school. You should find out how the rest of the world is."
-- Surely You're Joking Mr. Feynman
https://math.berkeley.edu/~stankova/
Not only she did grow in Bulgaria during the most turbolent times of regime change from communism to democracy, but later graduates with a PHD from Harvard, and later becomes Director and Founder of the Berkeley Math Circle, and is also organizer of math competitions in Bay Area, and publisher of what seems to be a complete set of Math Books, carefully crafted with her peers from BG and presented here
Curious whether miss Cairo was a student of hers or is to be.
https://sumizdat.com/homepage/index.html scroll down.
Besides how can she just skip college and go right into a PhD? She hasn’t even taken intro to Sociology or poetry yet.
Rather than speaking about her age or the vague notion of talent, I'd be much more interested in why the rest of the academia was unable to replicate her methods in 40 years or so. From her own admission, it was throwing different ideas and approaches until she saw a disprecancy that ultimately disproved the theorem. There should be many people far more experienced who do have the knowledge to do this, but why did they not? This dosen't look like a intuitive jump, seems more like building a test case that fully stressed the theorem.
Also, it's probably a not very well known conjecture that people wouldn't care too much about. Also also, it was a conjecture: people were probably looking to prove it rather that disprove it, failing and then giving up.
If I had to ask you kind sir, what would be the biggest life lesson (in coding, or anything general) that you could give me be?
Pursue what interests you and what fires up your passion, not what grown-ups tell you is “lucrative”, “prestigious”, “profitable” or somesuch.
So there is intel tiber cloud and it gave me a free jupyter and I wanted to play minecraft server, so I went to something which is a fork of it and I saw that there was a guy who had created rootless patch of ssh but it wasn't working for me so I kindly asked them and they were really kind enough to help me and then I used something like pinggy or the thousand other cloudflare tunnel-esque solutions which work without any problem whatsoever directly through ssh
Earlier I used to use junest to then get something like ssh working.
Can I be honest? I am so proud of myself because I guess I did all this stuff an year ago but I still remember the day I asked my friend to join the mc server and I joined too, the ping sucked but 8 gigs ram or even more and no storage limits and complete freedom... and I thought of such stuff after almost 2 weeks of trying different things and learned a lot about nat punching etc.
So such stuff always makes really sense to me, just tinkering with software. But I feel like I need to create some professional software too in order to leave a legacy or just, I feel like, if I am being honest, that the things that I am doing aren't that valuable. Anybody can learn such stuff from me from some blog post I write but there is nothing that I can ever monetize.
I don't know why I am seeking monetization that much but its just that, I see all these kids hustling my age too and I feel like everyone's doing everything for the money... so why should I be any different. I get a lot of ideas but I feel like I can never earn a single cent out of it and I find it a little frustating. I feel less confident myself regarding if I will ever earn decent money or will just be a low level cog of this whole system.
Found it https://github.com/mkj/dropbear I really love this project so much, it really helped me do something really wonderful.
* "Slope beats y-intercept." The best computer scientists and engineers I've ever worked with and/or mentored embodied this principle more than anything else.
* It can be tempting to over-optimize for short-term milestones (e.g., an important admissions exam, the next job or promotion), but there is a significant compounding value to knowledge accumulation and truly learning your craft well. Read and learn as much as you can, all the time, even if it isn't immediately necessary or useful.
Even in Scala (which is a very advanced language, but still far behind Idris) I deliberately don't use certain type-level features because it increases the compilation times too much (even with incremental compile!). It's very sad, but this is reality.
So, the problem isn't really that we need to "invent airplanes" - we already have them! What we need to do is make them usable and affordable by everyone.
I see more and more languages trying to add type-level features or try to embed other languages like Prolog-like ones into them. I hope that gets traction and becomes ergonomic, otherwise no one will use it in practice.
(He was publishing papers in high school.)
this is what gifted looks like, not legions of students who are told they are "gifted" but are just pretty smart
Be actually gifted and apply to a university that isn't run by dickheads.
Likely, there are also rule systems that would need to be circumvented; for example, while the admissions program for a graduate program would accept the student, the registrar or other entity might then negate that acceptance due to standing policy. At that point, I wouldn't be surprised if it reached the vice president level of the university (VP of the School that housed the department).
Every university handles this slightly differently. Exceptions can often be made.
To qualify you would normally have to do something truly exceptional or show a pattern of continuously excellent (grad student level). From what I can tell (I am not a math expert) she is considered a 'prodigy' and the problem she solved would require considerable skill, in a way that predicts continued successful performance at the university professor. The history of prodigies who skipped various steps of the normal schooling path is mixed- personally, I think people undervalue the importance of spending time with people in your age group during the 15-25 range, but honestly, without knowing more detail, it's hard to say.
Also, I love the handwritten slide on one of the photos. Very nice.
– were all homeschooled. Cairo started learning math using Khan Academy’s online lessons, and she quickly advanced through its standard curriculum. By the time she was 11 years old, she’d finished calculus
“There was this inescapable sameness, in a way. No matter what I did, I was in the same place doing mostly the same things,” she said. “I was very isolated, and nothing I could do could really change that. I’d wake up on certain days and realize, I’m just older.”
Wait, if everyone moves into the woods there will be no more woods that offer seclusion. Yeah moving into the woods is awful! I hope she doesn't burn out!
I don't have the knowledge to evaluate this, but I still think Cairo has not esta blished enough work to say she's far ahead of Kacyznski.
Jeez... what a damning indictment of today's Universities.
She could just use her publication as a dissertation and be done with it!
Many Master’s programs require Bachelor’s as a prerequisite for admission, and some of the others include a Bachelor’s at some point, but not all do. (The same is true of the Ph.D. and the stereotypically preceding degrees, and also of some professional degrees and Bachelor’s degrees.)
Was she not considered properly conditioned?
I know a few very special schools give undergrads access to brilliant minds in their field, but I also have been told that undergrads at those schools are mostly taught by grad students, so I'm not sure that Ivies provide a lot either, beyond the opportunity to hobnob with the legacies that will be running Goldman Sachs in 20 years.
Even at top universities, very very few freshmen are capable of doing high-quality research immediately. They'd be better served learning the foundations inside and out with a cohort of similarly strong students to challenge them.
Undergrads who do research generally aren't very good at research yet. A major reason is they either lack or don't fully understand the pre-reqs, which they progressively and cumulatively learn during undergrad. A student can be incredibly smart, but acquiring a strong rigorous math background will still take years.
Anyone can try doing research, even undergrads who half-know the foundations. However, trying research doesn't mean you have the background to do great research or to succeed in a postgrad program.
Plenty of people in postgrad programs don't know the foundations. It's ok. You are there to learn.
Completely unfair to expect someone already doing research to slog out 4yrs of classes not furthering their career.
I can see a committed and gifted student being able to get most of the pre-reqs for doctoral studies in America or Canada while in high school.
If you have bright enough undergrads, you change the curriculum for them within their field of expertise, so that they still get the breadth of things outside it while not wasting time with things they know. You let them not take as many classes, take graduate courses, do more research, take more courses from other departments in related areas but with different perspectives, and so on.
When I was an undergrad, in physics, there was a professor in the department who had done his undergrad there and was legendary, as was quietly mentioned in awe, for not taking any undergraduate physics courses while there; the department had let him skip all of them, and instead take graduate courses and do research.
My understanding is that US student loans accrue interest at rates above inflation. They do have repayment plans based on income, but because the interest rate is higher than wage growth, the debt just keeps growing if you're on these plans. They also don't have progressive thresholds like we do and the repayments tend to be higher.
Australia has a similar but less severe problem. Inflation and wage growth are closer together but they aren't the same. Still, the situation in the US seems worse.
We have no idea how many other people would achieve something similar with a similar background. Personally I'd bet almost anything it's a larger number than most people expect.
I'd also be surprised if she doesn't already have a pretty solid background in undergrad-level math.
The irony is she's actually more typical than not. Universities in the past were open to giving unusual talents special treatment.
Historically, the idea that everyone must follow the same path on the same timetable is unusual.
We are livestock for corporations, that's why.
There is no way this could be the “biggest flaw” in higher education today because the number of people possibly impacted is so tiny.
Although I think you’re striking at something that is a real problem with undergraduate degrees today: Many universities have become so watered down and softened that students spend the first 1-2 years doing what they should have been learning in high school.
My friends who still teach at university constantly complain about students arriving for undergrad with very poor writing, communication, and listening skills.
University (for folks serious about continuing in academia after) is (obviously) about making sure you have the same base knowledge as everyone else, but also for you to come to terms with how academia actually works, who the bad players are, who the good players are, and who you need to know to get shit to happen for you. So in that sense, most Universities going "no" is literally the most accurate reflection of what life's going to be like on a continuous basis on the inside.
And frankly I find it just as valuable.
There’s a modern phenomenon I’ve been thinking about but have struggled to put a name to.
Everything just becomes so generalized, streamlined that it becomes impossible to operate outside of the pre defined “happy path”.
AI will make this increase 100x as taking humans out of the loop seems to accelerate this process.
It is entirely possible for people to intensely focus on a very, very narrow thing - and ignore everything else. Even to such a degree that they can write a doctorate on it.
But I don't think that's a good excuse to make them forego other curriculum, especially if it is required for other students to take. Schools have a responsibility to educate people to a certain standard, and give them some general breadth.
The purpose of high school is to give you a wide foundation on everything.
The purpose of an undergraduate degree (in math) is to give you a wide foundation (in math).
In a (math) PhD, you are generally hyper-specialized in a very, very narrow area (of math).
Given her achievements, I'd be very surprised if Cairo hasn't already covered the material in an undergrad degree
If a young person is exceptional, do we force them into a liberal arts box? Surely there is value in literature and history. But this one young woman had found her passion. I have to believe that is she found out about something else, she would take that on.
Thankfully the workforce has common sense and will happily snap her up into employment.
Your own experience isn't generally applicable. Although it was a couple decades ago, I still use various things I learned from my non-major classes pretty much daily.
I don't think we need to nanny people who break the mold with extraordinary talents to conform to some generic correct educational sequence. They've proven they know how to make something of themselves and their own ideas should count for more.
I don’t know about you, but exposing these prodigies to some shared classes is not all that bad. Personally I think every student should be forced to take a class on ethics.
How many of the other students disproved 2 conjectures in advanced calculus that were open for decades?
sounds like Phd
I disagree that she should skip the general education.
The subjects such as English composition, she should be allowed to test out of if she is already a good writer.
This is a common STEM view, but it is inherently wrong. IMHO it dates back to the unfortunate divide between science and the liberal arts, whereas both were once considered a single field, now days there is disdain and mistrust between the two sides.
The point of history classes isn't to memorize the dates of wars, is it to understand the motivations of humans, it is to understand how the world we lived in has been shaped throughout time, and it is to learn how to do, and understand, research about the history of a place.
The point of English classes isn't just to get good at writing, it is to get good at various types of writing, it is to learn how to read different forms of literature, and it is to have a guided tour through a chosen selection of literature to hopefully develop one's character and thoughtfulness.
One of the most valuable classes I ever took at University was the Art Of Listening To Music. We started off around 500ad or so and went forward through time up until about 1920. We learned the vocabulary of music, how to sit down and listen to a piece of music and describe what we were hearing. After I was done with the class I went from appreciating a handfuls of genres of music to appreciating music itself no matter the genre. It was a 3 credit guaranteed A class that had enriched my life by an enormous amount.
If you really love your major, then the point of going to university was NOT your major, odds are you would've studied that field with or without the school. (Barring fields that require large capital investments, chemistry, physics, playing with an entire orchestra, building airplanes, etc) The point is everything else.
In reality, no one eats healthy food, everyone eats fast food hamburgers all day. Just look at the sales numbers of fast food / junk food VS organic lunch salad bars!
Except, that isn't true. Some people eat junk food all day, and some people choose to eat healthy. Obviously in America we have a bit of a bias, but just looking at averages doesn't give a complete picture.
> The ones that are known that the teacher doesn't take attendance are heavily desired.
Almost none of my university classes took attendance. Why would they? We were paying to be there, if we wanted to waste our money, it wasn't the university's problem.
> So most departments offer enormous 1 level classes with 200 students taught with minimal rigor
Reading books and writing essay's doesn't require rigor, the learning is in the doing. I put in honest work to learn and I got honest feedback from my 100 and 200 level professors, which was all I expected.
> And on top of that the frats all keep collections of graded tests from every class for years past
Almost none of my GE classes used multiple choice tests. They were typically essay tests, written in class.
I should note I did my GE requirements at a local community college, where class sizes averaged ~20-30 students, professors had office hours, and I think I only saw a TA once.
> This isn't education. This isn't worth six figures.
The price is too high yes, but a university is place to go where you can dedicate yourself fully to learning, hopefully w/o other outside worries (sky high tuition ruins that...). What a person chooses to do with that time is up to them.
Now one can argue that the worth of a degree is lessened by students who didn't actually learn all that much also being in possession of one. That is a closely related, but separate topic.
That said, the poster I was originally replying to was indirectly advocating for not caring about one's GE classes, and I was replying that one should indeed care, because those classes are incredibly important!
I also benefited tremendously from such a class. I would recommend it to anyone interested in music, or not interested in music and trying to figure out why others are interested in music.
Prodigy skips undergrad -> Universities are doomed? What?
The purpose of a PhD is not writing a dissertation. It is a research school, and I'm sure she could still learn a thing or two about research (and teaching).
> She could just use her publication as a dissertation and be done with it.
I’m not suggesting this person is doing anything fraudulent as she seems quite impressive.
However, educational institutions get constant requests from parents who want their children to skip far ahead before they’re ready. It’s a competitive world and they know that being able to claim a child skipped several grades or even skipped undergrad entirely is a unique and very impressive achievement for the resume. It also theoretically provides a few additional years of earning potential by giving a career head start.
The first problem is that many of these parents (again, no accusations for this specific case) see this and want to make it happen for their child at any cost. There are some wild stories about parents trying to cheat their kids forward or falsifying their accomplishments to try to skip grades.
The secondary problem is that it can be hard on kids to be thrust forward so far past their peers. I had several friends who skipped a grade in middle school and most of them didn’t have a great experience for social reasons. Skipping undergrad altogether would thrust someone into a foreign world with a lot of baseline expectations and norms that they hadn’t yet learned, combined with no peers their age to discuss it with.
It creates a high chance for burnout or failure, which could leave them worse off than when they started.
That’s why the recommendation is generally to do undergrad at a challenging institution that allows students some upward mobility in specific areas where they’re ahead. No reasonable undergrad program is going to have this person taking Algebra 101, but there are a lot of opportunities for them to jump right into advanced programs and go deep and broad.
Geometry required rote memory exceptionalism which also works against my brain design (adhd as well), but let me have the formulas to choose from and I'll get it done. Algebra II? I kept getting in trouble because I would do homework assignments in class instead of paying attention to the teacher trying to teach concepts that were easier for me to learn by reading the examples and following the book. After that, who would ever want to continue beyond the required credits and not think of further math as a masochist hellscape