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E.T. Bell called Fermat the Prince of Amateurs. One hundred years ago Ramanujan amazed the mathematical world. In between were many important amateurs and mathematicians off the beaten path, but what about the last one hundred years? Is it still possible for an amateur to make a significant contribution to mathematics? Can anyone cite examples of important works done by amateur mathematicians in the last one hundred years?

For a definition of amateur:

I think that to make the term "amateur" meaningful, it should mean someone who has had no formal instruction in mathematics past undergraduate school and does not maintain any sort of professional connection with mathematicians in the research world. – Harry Gindi

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    $\begingroup$ @Budney. Maybe not. It is a valid research question in history of mathematics. And not a trivial one. It has connotations to epistemology in mathematics. $\endgroup$
    – O.R.
    Commented Oct 30, 2010 at 14:49
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    $\begingroup$ There's a perfectly good question here: "What recent discoveries have amateur mathematicians made?" My advise to the OP: change the title to this, rather than the current title (which is what people seem to be arguing about, rather than the actual question), make it community wiki, and give it the tag "big list" - this applies even if the list is very short. $\endgroup$
    – Peter Shor
    Commented Oct 30, 2010 at 15:32
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    $\begingroup$ Part of the problem is: what is the definition of amateur? If it means someone who does mathematics but doesn't get paid for it, there are plenty out there who arguably make a significant contribution. $\endgroup$
    – Todd Trimble
    Commented Oct 30, 2010 at 16:00
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    $\begingroup$ Please make the appropriate changes, (e.g., those suggested by Peter Shor seem acceptable), and flag for moderator attention. $\endgroup$
    – S. Carnahan ♦
    Commented Oct 30, 2010 at 16:23
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    $\begingroup$ @Piertro: yes, E.T. Bell was an amateur historian of mathematics. :-) $\endgroup$
    – Todd Trimble
    Commented Oct 30, 2010 at 17:58

26 Answers 26

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About ten years ago Ahcène Lamari and Nicholas Buchdahl independently proved that all compact complex surfaces with even first Betti number are Kahler. This was known since 1983, but earlier proofs made use of the classification of surfaces to reduce to hard case-by-case verification.

At the time, Lamari was a teacher at a high school in Paris. Apparently he announced his result by crashing a conference in Paris and going up to Siu (who had proved the last case in the earlier proof in 1983) with a copy of his proof. Lamari's proof was published in the Annales de l'Institut Fourier in 1999 (Courants kählériens et surfaces compactes, Annales de l'institut Fourier, 49 no. 1 (1999), p. 263-285, doi:10.5802/aif.1673), next to Buchdahl's (On compact Kähler surfaces, Annales de l'institut Fourier, 49 no. 1 (1999), p. 287-302, doi: 10.5802/aif.1674)

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    $\begingroup$ Gunnar, it cool to know that Lamari was a teacher at that time (I was always a bit curious that there were two simultaneous proofs of the result). At the same time as I see according to mathgenealogy genealogy.ams.org/id.php?id=56039 Lamari accomplished his Phd at Paris VII in 1999. So, could we really call him an amateur...? $\endgroup$
    – Dmitri Panov
    Commented Apr 22, 2011 at 23:00
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    $\begingroup$ @Dmitri I'm told he was given a PhD for his proof. In any case, he got his PhD in 1999, after the result was published, so surely he still counted as an amateur? I've wondered about what happened to him afterwards... nothing turns up when I try to google him, so I wonder if he went back to teaching. $\endgroup$
    – Gunnar Þór Magnússon
    Commented Apr 23, 2011 at 8:47
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    $\begingroup$ it is harder than this. in the french system there is a a configuration, called "professeur aggregé", which is kind of wave-particle-dualism. if you are aggregé, you do hold a teacher position in a high school but can also spend some time at the university and may even decide to switch to academia again, later on. that said, and knowing a the french system a bit, i cannot imagine that someone can be awarded a ph.d. out of blue simply for doing something important. this might possibly happen in germany, say. in france there is a rigid procedure to become a ph.d. candidate in the first place. $\endgroup$
    – Delio Mugnolo
    Commented Nov 21, 2013 at 9:35
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    $\begingroup$ If you are paid to teach math, you are not an amateur, in any sense of this term, except maybe in a poetical sense. $\endgroup$
    – Patrick I-Z
    Commented Nov 21, 2013 at 14:31
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    $\begingroup$ @PatrickI-Z: That certainly depends. At least in Germany math teachers in school have the sad reputation of not knowing anything about mathematics "past undergraduate" level. (And this seems to be a phenomenon of the last few decades, the "old guard" of teachers seems to be more knowledgable). So in the sense of the question a lot of german school teachers would qualify as amateurs. $\endgroup$
    – Johannes Hahn
    Commented Nov 21, 2013 at 17:11
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After Martin Gardner published what one mathematician claimed to be a complete list of convex pentagons that could tile the plane, amateurs (Richard James III, a computer scientist, and Marjorie Rice, who had no mathematical training beyond high school) discovered several more classes of pentagons that could tile.

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    $\begingroup$ One place to read about this is Doris Schattschneider's contribution, In Praise Of Amateurs, to The Mathematical Gardner, edited by Klarner, 1981, pp 140-166. $\endgroup$
    – Gerry Myerson
    Commented Oct 30, 2010 at 21:36
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    $\begingroup$ When the Math Tower at Ohio State was built, the elevator lobby on each floor was given a tiling with a mathematical meaning. I believe the top floor exhibits one of Marjorie Rice's pentagonal tilings. As I recall, Ms Rice was invited to Columbus for the dedication of the building, but could not come for reasons of health. $\endgroup$
    – Gerald Edgar
    Commented Oct 30, 2010 at 21:49
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    $\begingroup$ More generally, Martin Gardner corresponded with countless amateur mathematicians who made significant contributions to recreational mathematics. Rice's achievements are probably the most spectacular, though. $\endgroup$
    – Timothy Chow
    Commented Oct 30, 2010 at 23:54
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    $\begingroup$ If you're ranking amateur mathematicians by their influence on mathematics in the last 100 years, rather than by the importance of the discoveries they made themselves, Martin Gardner is clearly on the top of the list. Among other things, I've met a number of mathematicians (myself included) for whom reading Gardner's Mathematical Games column as a kid was one of the things which led them to choose to go into mathematics. $\endgroup$
    – Peter Shor
    Commented Oct 31, 2010 at 14:32
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    $\begingroup$ I agree, I read parts of "The Complete Gardner" at least once a week, and reading his work was part of what persuaded me to choose to study math versus computer science. I already read Paulos, Sagan, Feynman, etc., but I don't know of anyone alive today doing what Martin Gardner did. If there is someone, I would love to know. $\endgroup$
    – Eric Tressler
    Commented Nov 1, 2010 at 5:07
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Greg Egan. He's a very renowned science fiction writer who holds a bachelor degree in mathematics. He wrote, as a coauthor, 2 articles which were published in peer-reviewed journals, one of them is with John Baez. The first one was written when he was approximately 40 years old.

There's also more eccentric example of Andrew Beal, which is much more known in the world of poker. He made however one minor conjecture in number theory for whose proof or disproof he offers $100,000.

And there's also a list on wikipedia which might be worth going through.

Edit: (nov-2018) Some recent progress by G. Egan has been made with an anonymous 4chan-member, on a problem on permutations.

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    $\begingroup$ Egan works closely with Baez and Dan Christensen, so one might argue that he doesn't meet Harry Gindi's definition of amateur. But actually, I think this is a very good example and I don't think connections with professional mathematicians make someone a professional (rather than an amateur). $\endgroup$
    – Dan Ramras
    Commented Oct 30, 2010 at 21:30
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    $\begingroup$ I like some of the "primary vocations" given on that wikipedia page, e.g., Benjamin Franklin (founding father), and Abraham de Moivre (bon vivant). $\endgroup$
    – Gerry Myerson
    Commented Oct 30, 2010 at 21:48
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    $\begingroup$ One guy which is missing from the list is Napoleon, who proved my <a href="mathworld.wolfram.com/NapoleonsTheorem.html">my favorite theorem in geometry</a>. $\endgroup$
    – Łukasz Grabowski
    Commented Oct 30, 2010 at 21:55
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    $\begingroup$ Lukasz, look again, Napoleon is definitely there. $\endgroup$
    – Gerry Myerson
    Commented Oct 31, 2010 at 5:26
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    $\begingroup$ Igor, Dan specifically mentioned Harry Gindi's definition, which is highlighted in the question statement. It includes the phrase, "does not maintain any sort of professional connection with mathematicians in the research world." I don't think Dan was minimizing Egan's contribution, I think he was pointing out a difficulty with Harry's definition. $\endgroup$
    – Gerry Myerson
    Commented Oct 31, 2010 at 10:39
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Aubrey de Grey, The chromatic number of the plane is at least $5$ (on the arXiv in April 2018)

Apparently, de Grey is a famous biogerontologist and he attacked the mathematical problem in his spare time.

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    $\begingroup$ See also this article in "Quanta": quantamagazine.org/… $\endgroup$
    – Sam Hopkins
    Commented Apr 20, 2018 at 12:33
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    $\begingroup$ And he has form for this. From what I’ve read, he had no formal training in biology when he started working on aging – though Cambridge subsequently granted him a PhD for his published work. Maybe he can send them this paper when it’s published, and get a PhD in mathematics too. $\endgroup$
    – Robin Houston
    Commented Apr 20, 2018 at 14:27
  • $\begingroup$ This has been published: Aubrey D. N. J. de Grey, The chromatic number of the plane is at least 5, Geombinatorics 28 (2018), no. 1, 18–31. $\endgroup$
    – YCor
    Commented Oct 20, 2022 at 7:01
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The complete graph on $n$ vertices $K_n$ is not planar for $n \geq 5.$ One may ask: what is the maximum Euler characteristic $\gamma(K_n)$ among all compact orientable surfaces into which $K_n$ may be embedded? It is a nice exercise to embed $K_5,$ $K_6,$ and $K_7$ into the torus. The final result was that $\gamma(K_n) = 2 \lfloor \frac{n (7 - n)}{12} \rfloor.$ In 1968 this theorem had been proven for "all cases except $n = 18,20,$ and $23.$ The proof was completed, at the end of the sixties, by Jean Mayer, a professor of French literature (!), when he found embeddings for these three values." (Surface topology, Firby and Gardiner, p. 111).

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  • $\begingroup$ A reference for this is the book "Map Coloring theorem" by Ringel and Youngs, which is curiously absent from the wikipedia article "Heawood conjecture" (an equivalent problem for $n>4$), which should also be corrected at least for confusing the division into n mod 12 cases with genus mod 12. $\endgroup$
    – BS.
    Commented Oct 17, 2020 at 14:35
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Kenneth A. Perko Jr. is a lawyer and an amateur topologist (with graduate-level training). In 1974 he found that two knots that were listed as separate knots in C. N. Little's "On knots, with a census for order 10" (1885) and similar tables, were actually identical.

Mathoverflow-user Daniel Moskovich recounted earlier on this site:

Little (with Tait and Kirkman) compiled his tables combinatorially. He drew all possible 4-valent graphs with some number of vertices (in this case 10), and resolved 4-valent vertices into crossings in all possible ways. He ended up with 210 knots. Then he worked BY HAND to eliminate doubles, by making physical models with string. He failed to bring these two knots to the same position, and concluded that they must be different. It took almost 100 years to find the ambient isotopy which shows that there are the same knot.

The book "Knots and Links" by Dale Rolfsen, published two years after Perko's publication, still lists the knots as different, they are knots [; 10_{161} ;] and [; 10_{162} ;] in Appendix C.

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    $\begingroup$ Unfortunately that's not completely accurate- the history isn't really that interesting. Perko was a student of Ralph Fox at Princeton. He left math without getting his PhD (although his 1964 senior thesis was quite important), and became a lawyer. 10 years later, in his free time, he messed around with math and did some research. He has 6 papers listed on MathSciNet, all very important, and all post-1974. He could have been a complete amateur to discover the Perko pair- but he wasn't. $\endgroup$
    – Daniel Moskovich
    Commented Dec 31, 2010 at 18:34
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    $\begingroup$ We have the following suggested edit (part 1): (The "rope" story is pure myth. In 1973, while I was completing the classification of 10 crossing knots, the duplication turned up quite naturally as the only undistinguishable pair. So I sketched some knot diagrams on a yellow legal pad and found out why. Sorry to disappoint, but Moskovich got it right. I was taking graduate math courses my last two years as a Princeton undergraduate, taught by the world's top knot theory topologists. $\endgroup$
    – S. Carnahan ♦
    Commented Oct 15, 2013 at 13:15
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    $\begingroup$ (part 2) Also, your second drawing is wrong. It's actually the mirror image of Rolfsen's 10-163, which some overly fastidious knot tabulators have re-named 10-162, adding a dose of confusion to an already difficult subject. --Ken Perko, October 12, 2013) $\endgroup$
    – S. Carnahan ♦
    Commented Oct 15, 2013 at 13:16
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    $\begingroup$ (part 3): Contrary to misinformation appearing elsewhere on Mathoverflow, I am not now, nor have I ever been, a Ph.D. --Ken Perko $\endgroup$
    – Carlo Beenakker
    Commented Oct 27, 2013 at 14:05
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An anonymous poster of a 4chan messaging board, in thinking about how long it would take to watch a 14-episode nonlinear anime program in any order, improved the lower bound for a length of a superpermutation. A superpermutation is a string that contains each permutation of $n$ elements as a substring. See OEIS A180632. Superpermutations are somewhat similar to De Bruijn sequences.

Whether the anonymous poster meets the definition of "amateur" may never be known, but the posting was from 2011, and apparently was noted by a handful of other mathematicians who think about these things not long afterwards.

The story has taken off in the public recently in part because Greg Egan, who was previously mentioned, has also in October 2018 improved the upper bound on the length of a minimal superpermutation.

Quanta Magazine has a nice article as well.

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    $\begingroup$ In a similar vein, the finishing touches on the proof that BB(5) = 47,176,870 (the 5th busy beaver number) were contributed by an anonymous Coq user called mxdys. $\endgroup$
    – Timothy Chow
    Commented Jul 15 at 16:10
  • $\begingroup$ @TimothyChow wouldn't that be a separate answer? $\endgroup$
    – Mark S
    Commented Jul 15 at 16:12
  • $\begingroup$ Maybe, but there are some arguable points. Is mxdys really an amateur? Is the contribution a "discovery" or rather a formalization of arguments that were already basically known and just not properly written up? $\endgroup$
    – Timothy Chow
    Commented Jul 15 at 18:43
  • $\begingroup$ Certainly! Terrance Tao has noticed the “comparative advantage” that amateurs may bring. $\endgroup$
    – Mark S
    Commented Aug 7 at 11:56
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Eugène Ehrhart was a high school teacher when he discovered the so-called Ehrhart polynomial, at the age of 55. He got his PhD at the age of 60.

Let $\Delta$ be a polytope with integral vertices in $\bf R^d$. Then there exists a polynomial $P$ such that for every integer $n$, $P(n)$ is the number of integral points in $n\Delta $. This polynomial satisfies a duality property: $(-1)^dP(-n)$ is the number of integral vertices in the interior of $P$. This duality property has been interpreted as Serre duality on toric varieties by Khovanskii in the 80's. The geometric interpretation of the coefficients of $P$ is still an open problem despite a huge literature. See http://icps.u-strasbg.fr/~clauss/Ehrhart.html for a short bibliography, and https://en.wikipedia.org/wiki/Ehrhart_polynomial for an introduction to the subject.

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  • $\begingroup$ You mean $n\Delta$ rather than $nP$? (And similarly about the interior.) $\endgroup$
    – Yaakov Baruch
    Commented Oct 27, 2018 at 16:54
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While this is on the front page again, I wanted to make mention of Joan Taylor, who discovered an aperiodic single tile, which she published with Joshua Socolar of Duke University in 2010. This is her bio blurb as it appears on their article in The Mathematical Intelligencer:

JOAN M. TAYLOR took up mathematics in 1991 at age 34 after being inspired by a magazine article on quasicrystals featuring Penrose’s rhombus tiling. She began but did not complete a degree, preferring to conduct her own research. Since then she has pursued tiling and related topics in abstract algebra and number theory including original work on constructible polygons. She likes to unwind with knitting and reading.

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There are many interesting discoveries made by mathematical distributed computing projects.

Their discoveries don't have an impact in the same way that theorems do, but from time to time resolving a theorem boils down to computation, and most of the participants are probably interested amateurs.

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    $\begingroup$ @Eric : This is a nice answer in a direction I hadn't thought of, thanks! What is Ramsey@Home about? Trying to improve the known bounds for small Ramsey numbers? $\endgroup$
    – Andrés E. Caicedo
    Commented Oct 30, 2010 at 19:36
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    $\begingroup$ @Andres: Yes. Quote from the site: "Ramsey@Home hopes to raise the lower bounds of R(3,3,3,3,3) or R(4,4,4,4,4)." $\endgroup$
    – Harun Šiljak
    Commented Oct 30, 2010 at 20:43
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    $\begingroup$ Is a computer an amateur mathematician? $\endgroup$
    – darij grinberg
    Commented Oct 31, 2010 at 18:57
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    $\begingroup$ @darji: phil.stackexchange.com $\endgroup$
    – Eric Tressler
    Commented Nov 1, 2010 at 4:24
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    $\begingroup$ Along this line it is also interesting to mention Riecoin, a decentralized virtual currency that broke the record on November 17, 2014 for the largest prime number sextuplet: riecoin.org/Press%20release%202014-11-21.pdf $\endgroup$
    – Tadashi
    Commented Jan 6, 2015 at 15:18
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Bill Gates co-authored the following paper in the 1970s with Christos Papadimitriou:

"Bounds for sorting by prefix reversal," Discrete Mathematics 27 (1979), no. 1, 47–57, MR0534952.

Not sure if Gates counts as an amateur, but he is at least a college dropout. :)

The only reason I know this is because once I ran across a book or article that discusses the results in this paper and then says something like, "Yes, this is THE Bill Gates." I was almost certain the book or article was by Knuth, but now I can't find the reference in any of my Knuth books. If someone else knows the reference I'm talking about, I would be grateful if they would post it as a comment to my answer. (It now bothers me that I can't find that reference. :) )

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  • $\begingroup$ Mike, would it be hofprints.hofstra.edu/43 , or jstor.org/pss/10.4169/194762110X489242 ? $\endgroup$
    – J. M. isn't a mathematician
    Commented Sep 4, 2011 at 5:10
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    $\begingroup$ @J.M.: No, although that paper pretty much says the same thing that the one I'm thinking of does. The reference I remember was longer ago than February 2010. Thanks anyway. :) $\endgroup$
    – Mike Spivey
    Commented Sep 5, 2011 at 20:48
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    $\begingroup$ Ah yes, Bill Gates' only contribution to computer science. $\endgroup$
    – JeffE
    Commented May 13, 2012 at 18:52
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    $\begingroup$ to JeffE : You mean the only positive contribution. $\endgroup$
    – Jérôme JEAN-CHARLES
    Commented Mar 9, 2016 at 0:16
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I think Escher qualifies. See Doris Schattschneider, The mathematical side of M. C. Escher, Notices of the American Mathematical Society 57 (2010) 706-718, http://www.ams.org/notices/201006/rtx100600706p.pdf

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Robert Ammann had some extremely important contributions to the study of aperiodic tilings, and to Quasi-crystals.

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In 2017, a counterexample to Conway's $1,000 conjecture “Climb to a Prime” was discovered by James Davis, “not a mathematician by any stretch”. (details)

Also in 2017, Dmitry Zakharov (then a 9th grader!) published a new bound for the 1962 problem of Danzer and Grünbaum, particularly improving a result of Erdős. His proof is elementary and takes just half a page!

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The important artists Anthony Hill and John Ernest proposed an upper bound for the crossing number of complete graphs (published by Richard Guy in 1960). Hill made other contributions to graph theory and was elected to the London Mathematical Society in 1979.

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    $\begingroup$ I just noticed this answer, and remembered seeing Hill's name in the acknowledgments of a paper by Dov Tamari ("A graphic theory of associativity and wordchain patterns", doi.org/10.1007/BFb0063002): "to Mr. A. Hill, an artist in London, for showing me some (completely) asymetric graphs and polyhedra." $\endgroup$
    – Noam Zeilberger
    Commented Nov 13, 2018 at 20:35
  • $\begingroup$ see books.google.co.uk/… $\endgroup$
    – Noam Zeilberger
    Commented Nov 13, 2018 at 20:36
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Kurt Heegner was a radio engineer by trade, but gave (essentially) the first proof of the Gauss class number one problem in 1952: namely that $\mathbb{Q}\sqrt{d}$ has class number $1$ if and only if $d \in \{-1, -2, -3, -7, -11, -19, -43, -67, -163\}$. Unfortunately, his work was largely ignored until around 1967, two years after his death.

His ideas also led to the development of Heegner points, which are very influential in modern number theory.

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    $\begingroup$ While it is true that Heegner did his PhD on radio engineering (after studying math and physics in Berlin) and also continued working in this direction, he already started publishing in math in 1929. Beginning with 1932, he could live from money coming in from his patents and, as I understand it, devoted his time mainly to math, and got in 1939 his habilitation. 1947-1950 he also worked for Zentralblatt (after he spoke with Schmidt and Hasse about his work on the class number one problem). [All information from the German wikipedia] So it seems like a peculiar borderline case of an amateur... $\endgroup$
    – Lennart Meier
    Commented Oct 28, 2018 at 6:46
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I was hoping that someone would add David Smith to this list. But a few days has passed and no one has brought this up. So please allow me to kick this thread a bit.

As discussed in another answer, it is recently discovered that the “einstein” tiling exists, settling a long-standing open problem. This discovery is primarily credited to David Smith, who describes himself as a shape hobbyist.

The story is also featured in QuantaMagazine.

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The composer Kieren MacMillan has written a few papers (see his postings on the arXiv) which prove some identities involving power sums by elementary means.

I suppose that it is in collaboration with a professional mathematician, but I think that he reasonably still counts as an amateur.

(Full disclosure: He is also a friend of mine)

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An old problem of Lebesgue circa 1914 asks what convex shape of smallest area can cover any planar set of diameter one. Upper bounds on the area of such a set have recently been improved to $a\leq .8440935944$ using elementary methods by Philip Gibbs in October of 2018. The pre-print is here and a Quanta article about the recent developments can be read here.

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    $\begingroup$ For what it's worth, it's an improvement of $2\times10^{-5}$. $\endgroup$
    – Fan Zheng
    Commented Jan 2, 2019 at 2:37
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    $\begingroup$ See also Baez's comments on this thread, with more on the history of improvements to this problem. Greg Egan, mentioned in two other answers already, has also helped Baez, Bagdasryan, and Gibbs with Mathematica verification of a previous improvement. $\endgroup$
    – Mark S
    Commented Jan 6, 2019 at 14:41
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Richard Friedberg, then an undergraduate pre-medical student, independently solved Post's problem (of whether there are intermediate Turing degrees) by the priority-injury method. This was a significant open problem at the time, so the result made news:

In Gödel's now famous letter to von Neumann that introduced the P vs NP problem, Gödel wrote

I do not know if you have heard that “Post’s problem”, whether there are degrees of unsolvability among problems of the form (∃y)φ(y, x), where φ is recursive, has been solved in the positive sense by a very young man by the name of Richard Friedberg. The solution is very elegant. Unfortunately, Friedberg does not intend to study mathematics, but rather medicine (apparently under the influence of his father).

Friedberg ended up becoming a physicist (Wikipedia biography).

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How about Saul Kripke?

Kripke-Platek set theory "is used all over the place, in recursion theory and set theory, [b]oth in classical results, and in fairly recent ones."

Modern philosophers (of science, mathematics, language etc. -- analytical philosophers) are probably a rich source of the list you seek. many do not have above-undergrad training in math, although I would use the definition of 'amateur' that we think of when we think of the ancients. that is, people who are distinctly in another field but make contributions to mathematics as part of their work or hobby.

As for other philosophers/logicians (off the top):

  1. Putnam
  2. Frank Ramsey (I don't have the rep to post more links--had a bunch for this question)

I know that by the time we get to someone like Ramsey, everyone's like: "surely that's not an amateur mathematician" but by the definition given in the question, i think he fits.

At any rate, you can probably find the names you're looking for in analytical philosophy, (mathematical/computational) economics/biology/linguistics, and so on.

The problem, I suspect, will be (in addition to the definition of 'amateur' which is not too difficult in my opinion, as long as you are satisfied with it for your list) the definitions of 'important' and 'discovery'. For example, I've known about Kripke's contributions for a while but I don't know, even now, whether this community considers them as important.

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    $\begingroup$ I would call Kripke's contributions to mathematical logic important (for what it's worth), but of course he is anything but an amateur. Rather, he is a professional from a closely related field. So I think your answer really addresses a different question, namely: "What are the important contributions of physicists, philosophers, computer scientists etc. etc. to mathematics?" $\endgroup$
    – Christian Remling
    Commented Jun 10, 2014 at 18:06
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There was a nice bit of mathematical modeling carried out by a professor of Law, John F. Banzhaf III.

He (re)created a power index in voting, which is better discussed on its wikipage: Banzhaf power index.

For Banzhaf's own work, see his paper:

Banzhaf III, J. F. (1964). Weighted voting doesn't work: A mathematical analysis. Rutgers L. Rev., 19, 317.

Incidentally, his particular approach was first formulated by Penrose (1946), then by Banzhaf (1965), and then yet again by Coleman (1971), thereby making it a potential candidate for the MO post about re-discovered (and re-published) works.

The mathematics around voting (the pinnacle being, perhaps, Arrow's impossibility theorem) provides other examples of amateur mathematicians at work. Consider, in particular, approaches to apportionment due to United States politicians John Quincy Adams, Alexander Hamilton, Thomas Jefferson, and Daniel Webster. (I omit further discussion of these latter examples, since they are arguably not "recent.")

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I don't know if this really qualifies, but I would say that Scott Draves can be viewed as an amateur mathematician for inventing/discovering the fractals known as Flame fractals. His work is more towards art, but there is a decent amount of math behind to optimize the aesthetically properties of the fractals. (The "nicest" fractal dimension is 1.52, for example).

Another more recent contribution regards superpermutations, by G. Egan and an anonymous 4chan user. See also this quanta magazine post.

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  • $\begingroup$ a link: en.wikipedia.org/wiki/Scott_Draves $\endgroup$
    – Artem Kaznatcheev
    Commented Nov 2, 2010 at 0:27
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    $\begingroup$ I'm looking at this thread (ordered by last activity), and one answer down is discussing superpermutations. $\endgroup$
    – Asaf Karagila ♦
    Commented Sep 25, 2020 at 20:41
  • $\begingroup$ @AsafKaragila Ah, I did not see that was posted already! $\endgroup$
    – Per Alexandersson
    Commented Sep 25, 2020 at 20:53
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The dentist Leon Bankoff, who died in 1997, published several mathematical papers. He has an Erdos number of 1. He has Wikipedia page. Here is a link to an interview of him by G.L. Alexanderson.

http://math.fau.edu/yiu/AEG2013/BankoffCMJ.pdf

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The American Institute of Mathematics, a nonprofit organization, was founded in 1994 by Silicon Valley businessmen John Fry and Steve Sorenson, longtime supporters of mathematical research.

That's Fry, as in Fry's Electronics, a retail chain in California.

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    $\begingroup$ Have Fry or Sorenson made any noteworthy mathematical contributions, in addition to their monetary contribution? $\endgroup$
    – Nate Eldredge
    Commented Nov 3, 2010 at 18:24
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    $\begingroup$ Beal, Escher, and Gardner are all on the list (with 10+ votes each), so I figured "impact on mathematics" qualified. $\endgroup$
    – Kevin O'Bryant
    Commented May 13, 2012 at 4:43
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In 2002, Manindra Agrawal, Neeraj Kayal, and Nitin Saxena proved the amazing result "PRIMES is in P," i.e., there is a deterministic, polynomial time algorithm for determining whether a given number is prime or not. The resulting paper was published in the Annals in 2004. The story was covered in the New York Times. The trio won the 2006 Fulkerson Prize, and the 2006 Gödel Prize.

In 2002, Kayal and Saxena were undergraduate students. I'd say that counts as "amateur."

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    $\begingroup$ I'd say it does not count as "amateur" at all. $\endgroup$
    – Fedor Petrov
    Commented Oct 16, 2020 at 19:16
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    $\begingroup$ @FedorPetrov You're kidding, right? The third highest answer is Greg Egan, who holds a Bachelor's degree in math and wrote a paper with John Baez (a famous research mathematician). These Indian students did not even have their Bachelor's degrees yet. And they made an extremely important discovery that rocked the theoretical CS world. $\endgroup$
    – David White
    Commented Oct 16, 2020 at 19:48
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    $\begingroup$ No, I do not. Math undergraduates often do important mathematical research (because many very clever people are sometimes undergraduates, and being at this stage they know already enough for making new contributions.) They do it much more often than former math undergraduates who already work in other fields (like Greg Egan at the age of 40). And working jointly with someone's students is also a very common thing. Of course, "PRIMES is in P" is an amazing work. But the authors were not amateurs, they were mathematicians at different stages of their careers. $\endgroup$
    – Fedor Petrov
    Commented Oct 16, 2020 at 20:36
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    $\begingroup$ I second @FedorPetrov's comments: "amateur" is not a synonym for "inexperienced", it is an (approximate) antonym of "professional" $\endgroup$
    – Yemon Choi
    Commented Oct 16, 2020 at 22:19
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    $\begingroup$ @YemonChoi By my reading, the students were not professionals, unless by "professional" one includes "intending to become a professional" (and how would one know that?). Actually, I am conflicted about calling Egan an "amateur" -- to me he walks and quacks just like a professional. $\endgroup$
    – Todd Trimble
    Commented Oct 17, 2020 at 16:18

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