10 ms·
At 17, Hannah Cairo solved a major math mystery
Related: https://news.ycombinator.com/item?id=44481441 https://news.ycombinator.com/item?id=44481441 - July 2025 (105 comments)
- shusaku 1y agoI feel like a constructive proof is the best scenario for a young talent like this, they really can use their vivid imagination and manipulate what they are after.
- tocs3 1y agoEarlier discussion: https://news.ycombinator.com/item?id=44481441 https://news.ycombinator.com/item?id=44481441 I wish her the best in her coming career.
- dang 1y agoThanks! Macroexpanded: A 17-year-old teen refutes a mathematical conjecture proposed 40 years ago - https://news.ycombinator.com/item?id=44481441 https://news.ycombinator.com/item?id=44481441 - July 2025 (105 comments)
- OutOfHere 1y agoWhat is the general basis for skipping college and getting admission directly in a Ph.D. program? What does one have to generally do to qualify?
- ameliaquining 1y agoAFAIK there's no general procedure; it's just that the admissions committee can admit anyone whom they're convinced can do Ph.D.-level work, and that almost always involves getting an undergraduate degree first but in extremely exceptional cases it might not.
- joe_the_user 1y agoEssentially, you have to convince a math professor that you able to complete the Ph.D program. There are a variety of ways to do that. Solving a major open conjuncture is clearly one of them but just getting to know one or another professors can work (I went to school with the semi-famous Paul Lockhart who also did this).
- TMWNN 1y agoFor context, Stephen Wolfram dropped out of Eton to start at Oxford, then was admitted to Caltech's physics PhD before finishing his Oxford BA, receiving his doctorate at the age of 21. (He was publishing papers in high school.)
- oldpersonintx2 1y agobe actually gifted this is what gifted looks like, not legions of students who are told they are "gifted" but are just pretty smart
- masfuerte 1y agoYes, but it's not enough. Be actually gifted and apply to a university that isn't run by dickheads.
- pinewurst 1y agoI knew someone who was admitted to a Masters program at Stanford sans undergraduate degree. His employer at the time was a large donor.
- dekhn 1y agoThe basis would be getting a dean's approval (or lack of a denial). That is the level at which "Exception Handling" is normally done at a university (above professor, below vice president), and it would normally be the dean of the department that housed the graduate program (some grad programs span multiple departments, which makes it more complicated). 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.
- anonzzzies 1y agoAs a pupil of Dijkstra and seeing at least some rise in formal verification because of the modern tooling and as a follower of Lean (and Agda, Coq, Idris* etc), I hope it will be at least a strive to deliver parts of proofs in code verifiable form. More machine verifiable building blocks will lead to a bettering of everything.
- Imustaskforhelp 1y agoOfftopic, but I am 17 too just like Hannah Cairo but nothing too groundbreaking till now I suppose and it absolutely brings me delight that I can talk to somebody who was a pupil of Dijkstra, I have heard a lot about dijkstra's algorithm's and I had forgotten about it and so I searched it right now, but the only thing I knew is that it is pretty popular algorithm. 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?
- Timwi 1y ago> what would be the biggest life lesson (in coding, or anything general) Pursue what interests you and what fires up your passion, not what grown-ups tell you is “lucrative”, “prestigious”, “profitable” or somesuch.
- anonzzzies 1y agoI have nothing to add to that.
- Imustaskforhelp 1y agoI mean, my interests are just really niche in the sense that I like to glue projects and one of my favourite hobbies is to search if there is an open source replacement to something or if I can use open source things to do stuff 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 https://github.com/mkj/dropbear I really love this project so much, it really helped me do something really wonderful.
- hadlock 1y agoA less click-baity headline might be "17 year old Hannah Cairo has solved the Mizohata-Takeuchi conjecture"
- noqc 1y agoprobably add "in the negative", to that.
- 1024core 1y ago"17 year old Hannah Cairo has disproved the Mizohata-Takeuchi conjecture"
- sdenton4 1y agoIs it an important conjecture, or just something someone came up with last week?
- cgh 1y agoThis is answered in the article’s second paragraph.
- sdenton4 1y agoBut I don't know that from the proposed re-titling of the article. The importance of the conjecture is more salient for the general reader than the (uninformative) names of the people who conjectured in the first place.
- brutal_chaos_ 1y agoIt's loosely explained deep down in the article.
- Biganon 1y ago...which is not the HN title, so parent's point still stands
- AtlasBarfed 1y ago”Cairo applied to 10 graduate programs. Six rejected her because she didn’t have a college degree. Two admitted her, but then higher-ups in those universities’ administrations overrode those decisions." 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.
- kurthr 1y agoA lot of programs don't want to have to babysit a teenager no matter how talented they are. Some of the more prestigious programs both have some experience with it and extra staff to (give profs warm fuzzies) handle any issues that come up. I'd expect it depends a lot on her interests and the particular professors that study that at any particular institution. Even as a student, I'd be more interested in which professors at Johns Hopkins were accepting students, than which school.
- xandrius 1y agoAs if that was the reason. Also, I've seen a great deal of adult babies in academia, so let's not be ageist here.
- kelp_herder 1y agoI suspect the reason is not about babying her as much as it is a concern for her development, about whether she would be "missing out" on the opportunities for social and intellectual growth in college. This kind of thing is usually why well intentioned higher ed people put roadblocks in front of precocious young people.
- sperginator 1y ago[dead]
- amelius 1y agoWhat I'd like to know is how many 17 year olds failed to solve their math mystery, and chose a career in programming instead.
- geodel 1y agoAlmost all of them?
- 1659447091 1y agoI never got excited for math, thus didn't care much to start with. But then add the issues of high school math and that solidified it for me. Math word problems were the worst. With a dyslexic brain I would consistently read the important words as something different, thus correctly solving the wrong problem and being derided for it. 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
- kevinventullo 1y agoMany people who succeed in solving their math mystery still end up choosing a career in programming
- carabiner 1y agoI hope she doesn't burn out and move into the woods like Grothendieck, Kaczynski,
- cindyllm 1y ago[dead]
- gspencley 1y agoI have yet to move into the woods but, given my current levels of burnout, I think such a proposition is highly underrated. What's wrong with moving into the woods? 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!
- bee_rider 1y agoOne option could be to get a hammock and go program in a tree. That’s sort of like moving to the woods, but you don’t have to live there full time and you still get to program.
- laurent_du 1y agoKaczynski was really not on the same level. By several orders of magnitude.
- BizarroLand 1y agoI mean, force feed her massive doses of LSD and use elite psyops to incessantly harass her as a ̶ ̶p̶r̶a̶n̶k̶ science experiment and maybe
- dekhn 1y agoIt's generally well documented that Kacynski was a highly talented mathematician.
- ykonstant 1y ago
- 1024core 1y ago> Only the University of Maryland and Johns Hopkins University were willing to welcome her straight into a doctoral program. She’ll start at Maryland in the fall. When she finishes, it will be her first degree. Jeez... what a damning indictment of today's Universities. She could just use her publication as a dissertation and be done with it!
- rahkiin 1y agoNormally you have a master of science as well. And for that you require a bachelors. So to do it directly you need to do it all at once?
- morleytj 1y agoIn the US a doctorate usually doesn't require a master's, most people go straight from undergrad to phd.
- sdenton4 1y agoTo add slightly more detail: Most research math programs in the US are 5(ish) year 'combined' masters+phd programs. The first couple years are basically course work, seminars, finding your area and advisor, and then the rest is the actual research work. It's not uncommon to leave after the first couple years with a Master's degree.
- barry-cotter 1y agoI have a Master’s and never got a Bachelor’s. Universities can admit people more or less as they please, subject only to the very forgiving scrutiny of accreditation agencies or whatever government department officially supervises them.
- dragonwriter 1y ago> Normally you have a master of science as well. And for that you require a bachelors. 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.)
- conferza 1y agoWow, this is remarkable. So inspiring to read, even though I'm terrible at mathematics.
- impish9208 1y ago“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 finally have something in common with a math prodigy.
- revskill 1y agoLoneliness is maththematician best friend.
- munificent 1y agoThank God she found math instead of Factorio.
- sbennettmcleish 1y agothe factory must grow ;-)
- _benj 1y agoMaybe hot take… 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
- MathMonkeyMan 1y agoHere's a link to the paper on the arxiv: https://arxiv.org/abs/2502.06137 https://arxiv.org/abs/2502.06137
- shermantanktop 1y ago- moved between countries or first/second gen immigrant? check - 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.
- stockresearcher 1y ago> moved between countries or first/second gen immigrant? “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.
- coderatlarge 1y ago“After the spring semester ended, her family moved from Davis to Berkeley — her brother had decided to transfer there — and Cairo finally felt able to settle in.” fascinating that the family follows the kids’ educational steps.
- gosub100 1y agoI wonder what would have happened if she said "mom, dad, I want to be a fashion designer/chef/interior decorator." Undoubtedly, they would have supported her ambition and individuality..of course.
- fmbb 1y agoAre home schooled over-represented among mathematicians that solved major math mysteries? Wait that would not even prove anything. Are there no mathematicians who solved major math mysteries that went to school?
- shermantanktop 1y agoI don’t know about major math achievements- the n is pretty small- but homeschooled kids are overrepresented in geography bees, spelling bees, and other academic competitions at the high school level. https://readlion.com/homeschoolers-dominate-inland-northwest-spelling-bee-qualifying-for-national-contest/ 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.
- zahlman 1y agoPreviously: Hannah Cairo: 17-year-old teen refutes a math conjecture proposed 40 years ago (https://news.ycombinator.com/item?id=44481441 https://news.ycombinator.com/item?id=44481441)
- fnord77 1y agoWait, what software engineering jobs require you to move to the Bahamas?
- cgh 1y agoIn my experience, online gambling or banking.
- smithkl42 1y agoInsurance is a pretty common one. Lots of insurance companies have offices there - and if you're a top-notch dev, you can make a crap-ton of money as well.
- OldfieldFund 1y agoRich people offshore money there
- zavg 1y agoThis is the most impressive thing I've seen in years.
- myth_drannon 1y agoI find the Soviet idea of Math Circles so interesting and important. I bought books on the subject, but it's difficult to implement for your own children only. Nothing beats it like having an actual one, run by math teachers and in your city.
- jarvist 1y agoAbsolutely. Key for me was to invite children's friends (and family) along, host it in our house and make it a recurring weekly thing. The books (presumably you also have these from the MSRI's 'Mathematical Circles Library') are great, but week-to-week I've found the free online NRICH resources much more directly useful: https://nrich.maths.org/about-nrich https://nrich.maths.org/about-nrich
- AlanYx 1y agoIt's wonderful that Khan Academy played a role in enriching her early education. It's proving to be a solid resource across the spectrum of math ability.
- spiderxxxx 1y agoThere's so many photos of just her staring off into the distance, and only one photo with her presenting the actual thing that she's supposedly famous for. I don't get the point of just all these random photos of this girl.
- azornathogron 1y agoIt's not very easy to take photos of mathematics, and the girl is the central subject of the article anyway. What images would you expect to see?
- shutupnerd0000 1y ago[flagged]
- dang 1y agoPlease don't do this here.
- shutupnerd0000 1y agoDo what exactly? The parent asked why there’s so many photos of her staring into the distance. Whats your theory on why there’s so many photos?
- dang 1y agoPlease don't post unsubstantive comments, and especially not shallow genderwar comments.
- shutupnerd0000 1y agoI was just answering the question posed above. I don’t think I’ve broken any rules. It just seems like you’re uncomfortable with my opinion. The quote in your profile is very ironic right now: “…It follows that the aim of healthy living is not the direct elimination of conflict, which is possible only by forcible suppression of one or other of its antagonistic components, but the toleration of it…”
- debo_ 1y agoHer notes are so clear and so artfully wrought! I wonder if learning from online resources makes one naturally focus more on presentation. From the article: https://www.quantamagazine.org/wp-content/uploads/2025/08/HannahCairo-cr.ValeriePlesch-Screen-768x488.webp https://www.quantamagazine.org/wp-content/uploads/2025/08/Ha...
- jagged-chisel 1y agoLooks like a prepared presentation rather than notes.
- baq 1y agoIf all presentations were prepared like this I’d probably attend them
- jagged-chisel 1y agoIndeed. All presenters can take a lesson from this talented person.
- thrance 1y agoI studied math and knew a girl like that, spending an inordinate amount of times making sure to understand every single details of what we were taught, even going further when I could barely keep up. She would also make those pretty, colorful notes with visual interpretations and drawings in the margin. It takes a special form of love for the discipline. She just finished her PhD and is now starting a postdoctoral contract.
- larodi 1y agoZvezdalina Stankova who comments on miss Cairo is on her own super out of the ordinary. https://math.berkeley.edu/~stankova/ 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 https://archi-math.com/ https://archi-math.com/ Curious whether miss Cairo was a student of hers or is to be.
- throwaway81523 1y agoAlexander Givental is also involved with that, I think. https://sumizdat.com/homepage/index.html https://sumizdat.com/homepage/index.html scroll down.
- rossant 1y agoAmazing story on a no-less amazing teenager. Also, I love the handwritten slide on one of the photos. Very nice.
- yumraj 1y agoWonder why Berkeley didn’t offer her PhD admission given that she was already working with the Prof there.
- terminalbraid 1y agoI've been at institutions that explicitly discourage that because of that reason. They want you to get breadth of experience rather than work with the same one person for your early career.
- IvyMike 1y ago> When I was an undergraduate at MIT I loved it. I thought it was a great place, and I wanted to go to graduate school there too, of course. But when I went to Professor Slater and told him of my intentions, he said, "We won’t let you in here." > 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
- sperginator 1y ago[dead]
- diimdeep 1y ago– 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 No normal childhood and certain kinds of parents, disproportionately more math education, there is certain downsides to all this, that articles like this conveniently avoid to mention.
- thrance 1y ago“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 don't think they're omitting it, but yeah. A lot of homeschooled kids don't fare nearly as well.
- corimaith 1y ago>Cairo kept reading and thinking. Eventually, she found a way to construct a strange, complicated function out of waves whose frequencies all lay on a curved surface — the type of surface the conjecture required. Usually, when you add these kinds of waves together, they interfere, canceling each other out in some places and reinforcing each other elsewhere. 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.
- thrance 1y agoHeh, take a look at her paper [0]. It is much more complex that TFA makes it out to be: once you have the intuition you need to prove it actually disproves 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. [0] https://arxiv.org/abs/2502.06137 https://arxiv.org/abs/2502.06137
- PilotJeff 1y agoThis math stuff is corrupting our youth. Besides how can she just skip college and go right into a PhD? She hasn’t even taken intro to Sociology or poetry yet.