# Research done by high-school students

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
• @Samurai, I haven't found her paper yet. – Joel Reyes Noche Jan 17 '15 at 3:31

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

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.

• 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 1:44
• 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. – Jorge Fernández Hidalgo Jul 23 '12 at 1:52
• 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 2:03
• 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
• @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

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.

• Galois is my hero ! – Fawzy Hegab Feb 4 '13 at 2:44
• Since you did mention about Galois what about Abel? – user210387 Jul 21 '15 at 13:40

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.

• This link seems relevant: dx.doi.org/10.1016/S0019-9958(67)90835-2 – Joel Reyes Noche Jul 24 '12 at 10:31
• the website at 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? – Joel Reyes Noche Jul 24 '12 at 10:35
• @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
• @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
• This is a famous (and rare) example of an important discovery made by a high school student. – zyx Feb 14 '13 at 21:12

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.

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.

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.

• 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

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

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.

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.

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.