While extremely talented, I am not surprised to find this coming from a teen. Major mathematical discoveries often have come from those in their mid 20’s with the greater discoveries being skewed towards the younger 20s and teens. I think this because pure mathematics is just so creative.
paulpauper · 2h ago
Trying to do anything original and novel in math is extremely hard at any age. to do it at 17 is insanely talented. congrats
No comments yet
ledauphin · 2h ago
here's a dumb question:
she's starting her Ph.D. this fall - hasn't she already achieved it? What is the theory behind expecting someone who has solved a decades-old problem to do some "second" thing to prove that they have extended the bounds of human knowledge?
EvgeniyZh · 1h ago
Ph.D. is training in how to do research. Solving one, even very hard problem not necessarily means that you don't need such training. It's especially tricky with counterexamples which sometimes question of raw talent and luck rather than skill.
The next step for someone who has PhD and want to stay in academia is postdoc. After solving one problem, you would not necessarily have what's needed to get a good postdoc, such as clear research agenda or proof of ability to publish consistently.
daxfohl · 37m ago
A PhD is as much a stamp of endurance as it is a stamp of intelligence or accomplishment.
parpfish · 1h ago
But what does somebody do with a PhD at age 17? I can’t imagine hiring them as a prof when they’re so young. It’s not a bad idea to just take a couple years to continue your already productive collaboration while getting mentored on the non-math parts of being a mathematician.
ics · 1h ago
IIRC Erik Demaine (https://en.wikipedia.org/wiki/Erik_Demaine) started teaching at 20 and had his PhD. I can't remember if I first saw his name because of the MacArthur Grant or one of those science documentaries but one of his pages was on the frontpage here a week or two ago and it seems like he's been thriving.
tehjoker · 4m ago
When she graduates she'll probably be between 20 to 23 years old.
nextos · 2h ago
A PhD in the US requires a lot of coursework, aside from research. Perhaps, she is interested in that. Otherwise, some universities, especially in EU, offer PhDs by publication. She could simply wrap up her counter-example publication (https://arxiv.org/pdf/2502.06137) as a thesis and possibly graduate. Sometimes, you can even do this without a supervisor.
xg15 · 2h ago
Sounds as if she even has a potential supervisor:
> “It took me a while to convince Ruixiang Zhang [the professor of the course where the problem had been posed] that my proposal was actually correct,” Cairo says
> At the University of Maryland, she will continue working under the supervision of Zhang. “He helped me so much, and I’m really grateful. Beyond his class, which I loved, he spent countless hours tutoring me,” she recalls.
pclmulqdq · 51m ago
PhD by publication usually takes a bit more work. I think they tend to want 3 related papers in a field.
almostgotcaught · 32m ago
That's a rule of thumb for applied sciences. Plenty of theory PhDs graduate with 1 or 0 papers.
pclmulqdq · 28m ago
Nobody gets a PhD by publication with 0 publications. This is usually a backdoor for people who have done a lot of work in a field, certainly far more than a PhD thesis, and have just never gotten the credential.
stogot · 44m ago
What level or type of publication is required?
pclmulqdq · 22m ago
They must be peer-reviewed journal papers and I believe they tend to prefer if at least one is well-cited or significant, especially if you have only three papers. It is generally harder to get a PhD by publication than to get a PhD the normal way.
eviks · 1h ago
There is no deep theory here, bureaucracy doesn't think deep.
MPSFounder · 21m ago
Great question. I have a PhD. People forgot the purpose of a PhD. Hannah effectively achieved what many with a PhD fail to do, and that is contribute novel research. A PhD in the US (only place I can comment on) has lately been focused first and foremost on a) preparing for academia, which entails teaching and a lot of courses, and b) research for industry positions (many students in my cohort were from China or India and this was their segway into a job in the US). I agree a PhD should be purely focused on research and extending human knowledge. In practice, it is a business where students go to conferences to promote their PI's work, where Universities get cheap lecturers in the form of TAs, and where many mediocre students write incremental papers to secure an RnD position (change this by a little and see how it affects your results. This is your paper). I am very impressed by Hannah's work though and she embodies the selfless nature of research that is very much missing. I see too often people seeking to advance their own career and pick a PhD route of least resistance. While they are entitled to maximize profits, and oftentimes do not want to go to academia where solving the impossible is admired, we must remember discoveries often hinge on challenging problems and a selfless pursuit of the impossible. This is just my opinion based on what I saw in my cohort and at 30+ conferences
Keyframe · 2h ago
Original title is more informative than the edited one here.
leephillips · 2h ago
I submitted under an approximation of the original title, and it was edited within seconds.
miles · 1h ago
There is too much "helpful" title modification of late. The original title itself fits within HN limits:
"A 17-year-old teen refutes a mathematical conjecture proposed 40 years ago"
The site's guidelines are clear[1] but increasingly ignored by some moderators:
"...please use the original title, unless it is misleading or linkbait; don't editorialize."
As the submitter, you can re-edit the title after submission (for some limited time period).
tomjen3 · 2h ago
>One day, he proposed proving a special, much simpler case of the conjecture as a homework assignment. As an optional part, he included the original conjecture
There is a lesson there: always give people an opportunity to excel, if you can.
sshine · 30m ago
I remember at first year of university being presented with a bunch of “simple” problems early on, such as the Collatz conjecture.
I remember wanting to spend time trying to explore what a solution might look like, because such simply formulated problems must have equally simple solutions.
Maturing and getting a better understanding of my intellectual capacity, I have opted to solve practical problems with a much bigger chance of success and absolutely no groundbreaking qualities.
But I liked being taken serious from the start, and I think it’s important to try and solve hard problems before you grow stuck in the real world.
This sort of transform (what I think many people call inverse problems) is quite common in reconstruction problems- that is, where you pass light or other EM through an object, the light scatters, and hits a detector. Typically you want to find the minimum error reconstruction. See more here: https://en.wikipedia.org/wiki/Radon_transform
raincom · 1h ago
Great achievement. Now Princeton Math department will ask her to join their school for Ph.D.
kemitchell · 3h ago
Refuted?
zahlman · 2h ago
Yes, either proving a true conjecture or refuting a false one is "solving" it.
qsort · 2h ago
The Mizohata-Takeuchi conjecture is a statement in the form "For all <x> (a bunch of math)".
Showing that there exists an x such that the statement is false disproves the conjecture.
She found a counterexample.
mgiampapa · 2h ago
She found more than one way of disproving it in the process.
gilleain · 2h ago
Yes, found a counterexample to the conjecture.
old_man_cato · 3h ago
[flagged]
dang · 12m ago
Could you please stop posting unsubstantive comments to Hacker News? we're trying for something different here.
Also, Terence Tao hinted at some further advances some time ago [2], does anyone know more about that?
[1]: https://www.youtube.com/watch?v=3ZeH_8sTyKA
[2]: https://mathstodon.xyz/@tao/114003793236630744
https://terrytao.wordpress.com/2025/02/25/the-three-dimensio...
No comments yet
she's starting her Ph.D. this fall - hasn't she already achieved it? What is the theory behind expecting someone who has solved a decades-old problem to do some "second" thing to prove that they have extended the bounds of human knowledge?
The next step for someone who has PhD and want to stay in academia is postdoc. After solving one problem, you would not necessarily have what's needed to get a good postdoc, such as clear research agenda or proof of ability to publish consistently.
> “It took me a while to convince Ruixiang Zhang [the professor of the course where the problem had been posed] that my proposal was actually correct,” Cairo says
> At the University of Maryland, she will continue working under the supervision of Zhang. “He helped me so much, and I’m really grateful. Beyond his class, which I loved, he spent countless hours tutoring me,” she recalls.
"A 17-year-old teen refutes a mathematical conjecture proposed 40 years ago"
The site's guidelines are clear[1] but increasingly ignored by some moderators:
"...please use the original title, unless it is misleading or linkbait; don't editorialize."
[1] https://news.ycombinator.com/newsguidelines.html
There is a lesson there: always give people an opportunity to excel, if you can.
I remember wanting to spend time trying to explore what a solution might look like, because such simply formulated problems must have equally simple solutions.
Maturing and getting a better understanding of my intellectual capacity, I have opted to solve practical problems with a much bigger chance of success and absolutely no groundbreaking qualities.
But I liked being taken serious from the start, and I think it’s important to try and solve hard problems before you grow stuck in the real world.
https://arxiv.org/abs/2502.06137
I had the opportunity to take a harmonic analysis course in grad school. I passed it up. It was only tangentially related to my research at the time.
https://www.nytimes.com/interactive/2025/06/30/science/math-...
Showing that there exists an x such that the statement is false disproves the conjecture.
She found a counterexample.