Research done by high-school students

Question

I'm giving a talk soon to a group of high-school students about open problems in mathematics that high-school students could understand. To inspire them, I would like to give them examples of high-school students who have made original contributions in mathematics. One example I have is the 11th-grader from Hawai'i named Kang Ying Liu who in 2010 "discover[ed] nine new geometric formulas for describing triangle inequalities."

Do you have any other examples of high-school students who have made original contributions in mathematics?

Hello! Can you tell me where can I find the inequalities by Kang Ying Liu (if they are available for reading)? — Henry, Jan 16 '15 at 17:01
I don't recall whether Deligne was in high school in the story that when he attended a local university seminar where the question was posed whether every topos has enough points, Deligne was the one who was able to come up with a counterexample. — Daniel Schepler, Aug 03 '21 at 00:38

score 53 · Answer 1 · edited Sep 22 '16 at 16:26

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In 1988 in the IMO, Australia decided to use the following question:

Let $a$ and $b$ be positive integers such that $ab+1$ divides $a^2+b^2$. Prove $\frac{a^2+b^2}{ab+1}$ is a perfect square.

The problem was proposed by Stephan Beck, West Germany. No one in the committee was able to solve it. Two of it's members were George Szekeres (Erdős number of 1) and his wife, both famous problem solvers and problem creators. The problem was then sent to 4 prominent number theory researchers and they were asked to work on it for six hours. None of them could solve it in this time. The problem committee submitted it to the jury of 19th IMO marked with a double asterisk, which meant a super-hard problem, possibly too hard to pose. After a long discussion, the jury finally had the courage to choose it as the last problem of the competition.

Eleven students gave perfect solutions. The solution to the question used a new technique in problem solving that had never been used before. However 11 high school students were able to surpass prominent number theorists in their own field by solving the question. The technique used for solving the problem is called Vieta Jumping.

edited Sep 22 '16 at 16:26

SchrodingersCat

24,994

answered Jul 23 '12 at 01:37

Asinomás

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This doesn't really qualify as research. 6 hours is nothing to a research mathematician, and of course IMO contests are going to be primed to quickly find tricky elementary solutions. As this was an already solved contest question, I don't think it counts. – Potato Jul 23 '12 at 01:44
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Yes thats true but Vieta jumping had never been used before. Cristopher Columbus wasnt the first person to get to America but we recognize him as a discoverer anyways. Its still pretty amazing that those kids managed to do in one hour and a half what experts couldn't do in 6 hours. – Asinomás Jul 23 '12 at 01:52
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Except experts in number theory are generally concerned with things other than tricky elementary problems. And clearly it had been used before, because the IMO doesn't give unsolved problems. So someone had already solved it. – Potato Jul 23 '12 at 02:03
The answer got a negative vote, but I don't think that you should delete this answer. – Asaf Karagila Jul 23 '12 at 15:21
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Potato's strictures about the solution to the problem not being research brings to mind Richard Bellman's book Dynamic Programming in which at the end of each chapter is a section titled "Homework Exercises and Research Problems". When Bellman was asked how someone could tell them apart, he answered "If you can solve it, it is a homework exercise; if not, it is a research problem." – Dilip Sarwate Jul 23 '12 at 20:40
Was this excerpt taken by "Problem - Solving strategies", by engel? http://www.springer.com/mathematics/book/978-0-387-98219-9 – Ant Jul 01 '14 at 23:25
No it wasn't, I don't remember very well where I saw it, but I only discovered that book very recently, I like it a lot though. – Asinomás Jul 01 '14 at 23:32
Potato's comments seem off to me & seem to indicate a lack of familiarity with the field of number theory. presumably no experts volunteered to solve the problem after working on it for 6 hours, so the 6 hour limit is only a lower bound. the experts may not have thought it solvable. also number theory is rampant for problems that "look elementary" but proofs can be very difficult to find, even "elementary" ones. a famous example is finding primes between $n$ and $2n$ by Erdos. he found a new "elementary" proof that was celebrated. it maybe was not conjectured to even exist before he found it. – vzn Jan 08 '15 at 20:09
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@vzn Sure, "elementary" problems often lead to good mathematics. I'm not disputing that. What I am disputing is that solving problems for a math contest counts as research. Finding this "Vieta jumping" solution was a great mathematical feat, but it was not research. – Potato Jan 29 '15 at 21:53
not sure about your claim/ slightly skeptical re "the IMO doesnt give unsolved problems". is that official written policy anywhere? while thats plausible it seems not inconceivable they might throw in an occasional conjecture from somewhere esp if students are given partial credit for some real insights that fall short of solving the entire problem. agreed it is not typical research in various ways but one essence of research is to find a solution to a problem that was not previously known. fyi iirc this problem appeared in usenet sci.math around ~1989... – vzn Jan 29 '15 at 22:10

MJD · Accepted Answer · 2022-03-11T01:04:29.173

I'm not sure this is really what you're looking for, but Britney Gallivan, then $16$, disproved the famous claim that it was impossible to fold a piece of paper in half ten times, by folding one twelve times. She also came up with a model that correctly explained the limit, and predicted how big the original paper would have to be to be folded $n$ times.

Archive of page about Gallivan from the Pomona Historical Society

score 27 · Answer 3 · answered Jul 23 '12 at 15:58

27

He may not have been in "high school" but he was certainly at that age when he "was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals, thereby solving a long-standing problem." I'm of course talking about Galois, whom I am amazed has not been mentioned yet.

answered Jul 23 '12 at 15:58

Derek Allums

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Galois is my hero ! – FNH Feb 04 '13 at 02:44
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Since you did mention about Galois what about Abel? – Jul 21 '15 at 13:40

Dilip Sarwate · Answer 4 · 2012-07-24T16:22:01.067

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The Nordstrom-Robinson $(16,2^8,6)$ nonlinear binary code was discovered by A. W. Nordstrom, then a high-school student in Illinois, after J. P. Robinson, a faculty member at U. Iowa, gave a talk at the high school about unsolved problems in coding theory. It is the simplest example of nonlinear binary codes with more codewords than linear binary codes with the same minimum distance $6$. The generalization was discovered by F. P. Preparata.

edited Jul 24 '12 at 16:22

answered Jul 23 '12 at 00:34

Dilip Sarwate

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This link seems relevant: http://dx.doi.org/10.1016/S0019-9958(67)90835-2 – JRN Jul 24 '12 at 10:31
the website at http://www.sciencedirect.com/science/article/pii/S0019995867908352 states that "The research reported in this paper was supported by the National Science Foundation under the Research Participation Program for Exceptional Secondary Students and Grant GK-816." But aside from this, do you know of any written reference stating that Nordstrom was a high-school student when he made this discovery? – JRN Jul 24 '12 at 10:35
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@JoelReyesNoche I don't know of any written reference but the story was widely accepted in coding theory circles in the late 1960s (I heard it first in early 1969 from my professor in a coding theory course), and Robinson confirmed it to me during a conversation at a conference in 1973. He always gave credit to Nordstrom for the discovery and emphasized that it came about as a result of his (JPR's) talk at Nordstrom's high school. Presumably the talk was part of Robinson's work on the NSF grant. – Dilip Sarwate Jul 24 '12 at 13:25
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@JoelReyesNoche The link you provided lists Nordstrom's affiliation (on the published paper) as a high school in Moline, Illinois, just across the Misissipi River from Iowa. – Dilip Sarwate Jul 24 '12 at 16:20
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This is a famous (and rare) example of an important discovery made by a high school student. – zyx Feb 14 '13 at 21:12

score 15 · Answer 5 · answered Jul 23 '12 at 00:35

The then 16-year old Sarah Flannery published an algorithm for public-key cryptography that she dubbed the "Cayley-Purser algorithm". There was some excitement when it was found to be a bit faster than RSA, but was subsequently found to be flawed. Nevertheless, it was still quite an accomplishment for a teenager.

Sarah's paper can be seen here. She has written a book with her father on her experiences.

Dave L. Renfro · Answer 6 · 2014-09-18T16:37:43.047

Surprisingly (because it's so recent) missing is any mention of Jacob Lurie's research that earned him 1st place in the 1996 Westinghouse Science Talent Search. But perhaps not surprisingly, as others here may have felt it wasn't something the original poster could have used.

See the two articles about it in the July 1996 Notices of the American Mathematical Society.

In the 1st article John H. Conway wrote: His writing is already very much like that of a professional mathematician, and any professional mathematician would be justly proud if he could produce arguments as subtle and deep as Lurie's.

In the 2nd article Allyn Jackson wrote: Gasarch believes that Lurie's paper "could easily be half a Ph.D. thesis."

18 Sept. 2014 UPDATE: I just read that Jacob Lurie is among those selected this year (2014) for a MacArthur Fellowship.

score 12 · Answer 7 · answered Jul 23 '12 at 03:22

12

In 2003 the Kemnitz Conjecture, a 20 year old open problem in combinatorial number theory, was independently proven by Christian Reiher and Carlos di Fiore. Reiher at the time had just passed his Abitur (entrance exam for German universities), and di Fiore was still a high school student.

answered Jul 23 '12 at 03:22

Pete L. Clark

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It is likely that Reiher was in the extra thirteenth year of high school that exists in some parts of the German system. Those two students had 8 IMO participations and 5 gold medals between them, which is on the high side even for high school students who do independent research. – zyx Feb 14 '13 at 21:21

score 9 · Answer 8 · answered Jul 23 '12 at 00:24

You could look at the winners of the Intel Talent Search and projects at the Research Summer Insitute at MIT.

Some high schoolers have also written papers at Ken Ono's REU. You can find them by looking at the archive on his website. Here are two:

http://www.mathcs.emory.edu/~ono/REUs/archive/results/reu09FengKirschMcCallWage.pdf

http://www.mathcs.emory.edu/~ono/REUs/archive/results/reu09DummitGoldbergPerry.pdf

score 9 · Answer 9 · edited Jul 23 '12 at 16:01

9

Gauss anecdotally solved the finite summation of linear sequences in primary school. Although to be fair this wasn't an original discovery as there are demonstrations of pairwise summation as early as 400 C.E. in Jewish religious works.

edited Jul 23 '12 at 16:01

answered Jul 23 '12 at 08:34

jhoyla

101

Vincent Yu · Answer 10 · 2012-07-23T06:56:32.733

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The Siemens Competition is for high school students doing original research in math and science. They provide a list of the math paper abstracts of the winners and finalists.

As an example, here is the paper from 2009's winning project.

edited Jul 23 '12 at 06:56

answered Jul 23 '12 at 06:31

Vincent Yu

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score 5 · Answer 11 · answered Apr 08 '13 at 15:48

5

Sylvain Cappell's paper, "The Theory of Semi-cyclical Groups with Special Reference to Non-Aristotelian Logic," won him the 1963 Westinghouse Talent Search when he was 16 years old, as recollected in this Boy's Life article. This was the first of Sylvain's many important mathematical discoveries. Tangentially interesting is that the runner-up in this contest, Sylvain's rival, was a certain William Leonard Pickard.

answered Apr 08 '13 at 15:48

Daniel Moskovich

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Pickard was a finalist in that year, but not "the" runner up (i.e. 2nd place). That went to Arthur Shapiro. https://student.societyforscience.org/science-talent-search-1963 – Jacob C. Oct 09 '17 at 17:49

score 3 · Answer 12 · answered Aug 18 '25 at 16:12

A recent example that's currently^{(August 2025)} in the news would be Hannah Cairo.

It’s not that anyone ever said sophisticated math problems can’t be solved by teenagers who haven’t finished high school. But the odds of such a result would have seemed long.

Yet a paper posted on February 10 left the math world by turns stunned, delighted and ready to welcome a bold new talent into its midst. Its author was Hannah Cairo, just 17 at the time. She had solved a 40-year-old mystery about how functions behave, called the Mizohata-Takeuchi conjecture.

. . .

In 2021, amid the Covid pandemic, Cairo’s world began to widen, even as it was narrowing for so many others. Travel restrictions stranded her family at her grandparents’ house in Chicago. While they were there, she joined the Math Circles of Chicago, where teachers and students gather to solve difficult problems together.

The experience led her to apply the following year to a two-week online summer program run by the Berkeley Math Circle, which has nurtured some of the most exceptional math talent in the world. In her application, she listed a set of self-taught subjects that together comprised the equivalent of an advanced undergraduate math degree. She was 14 years old.

. . .

The math world is also adjusting to the fact of Cairo herself. After completing the proof, she decided to apply straight to graduate school, skipping college (and a high school diploma) altogether. As she saw it, she was already living the life of a graduate student. 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.

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.

score 1 · Answer 13 · answered Mar 19 '13 at 15:08

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The following link provides additional examples: http://isef.kpi.ua/index.php/tezy-proektiv/81-osnovna/pro-konkurs/tezy/96-tezy-isef-2011-ma

answered Mar 19 '13 at 15:08

JRN

6,731

score 1 · Answer 14 · answered Aug 30 '25 at 20:40

Not as prominent, but I've worked with high school students as part of a regular research seminar at the high school I teach at. They study a major topic in the fall semester (generally number theory or graph theory) and then we work on a research project. These are students who have finished multivariable calculus and linear algebra the year prior and are generally juniors and seniors. We've gotten published papers out 2 of the last six years we've done this. One of these papers was a graph theory paper published in Involve, which is a journal dedicated to undergrad research. See arxiv version here. The other paper was recently published in Integers and concerned 2-near perfect numbers. That paper is here(pdf).

Research done by high-school students

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