Did amateurs ever produce important proofs or similar?

Question

Background

Mathematics and some areas of physics and computer science have the peculiar appeal that some problems and results are easy to understand and it is conceivable that somebody armed with nothing but the right idea can come up with something groundbreaking. This makes these fields particularly prone to amateurs becoming obsessed with solving famous problems or debunking something established. Some of these erroneously think they succeeded and then proceed to pester professional scientists (and are typically called cranks or crackpots).

Now, there are often good arguments that it is unlikely that an amateur is actually onto something, but here I am wondering about the empirical side:

Question

What are the most relevant results (if any) produced by amateurs in these fields? Some specifications:

• Eligible results must be pen-and-paper theory, possibly aided by a computer.

• Finding a simpler or completely different proof for a solved problem is eligible.

• Results that can in principle be found by brute-force computing are not eligible (even if they require some search strategy to avoid combinatorial explosion). It’s not that such results are without value, but they do not fit what I am interested in for several reasons (results are easy to check; it’s more plausible to get lucky; obsession may be a virtue; …).

• For the purpose of this question, an amateur is somebody who never did any of the following:

  • acquire an academic degree in a field that features proofs (mathematics, physics, computer science, …),
  • publish a paper in one of these fields,
  • make a living of performing research in such a field.

• The result must have been found after 1960. This is not a hard deadline; I mainly want to ensure that the basics of mathematics and physics had been thoroughly explored and to somewhat ensure that there were no amateurs who would not be amateurs nowadays.

Answer 1

A case from this year is that of Aubrey de Grey.

Aubrey de Grey, a biologist known for his claims that people alive today will live to the age of 1,000, posted a paper to the scientific preprint site arxiv.org with the title “The Chromatic Number of the Plane Is at Least 5.” In it, he describes the construction of a unit-distance graph that can’t be colored with only four colors. The finding represents the first major advance in solving the problem since shortly after it was introduced. “I got extraordinarily lucky,” de Grey said. “It’s not every day that somebody comes up with the solution to a 60-year-old problem.”

De Grey appears to be an unlikely mathematical trailblazer. He is the co-founder and chief science officer of an organization that aims to develop technologies for “reversing the negative effects of aging.”

https://www.quantamagazine.org/decades-old-graph-problem-yields-to-amateur-mathematician-20180417/

The excellent Quanta Magazine likes to report about such exciting happenings in the world of science. They have covered those mentioned so far except Kurt Heegner:

• Marjorie Rice

• Yitang Zhang

Then there other cases of professionals who, like Yitang Zhang, count as outsiders:

As he was brushing his teeth on the morning of July 17, 2014, Thomas Royen, a little-known retired German statistician, suddenly lit upon the proof of a famous conjecture at the intersection of geometry, probability theory and statistics that had eluded top experts for decades.

Known as the Gaussian correlation inequality (GCI), the conjecture originated in the 1950s, was posed in its most elegant form in 1972 and has held mathematicians in its thrall ever since.

https://www.quantamagazine.org/statistician-proves-gaussian-correlation-inequality-20170328/

Answer 2

A well-known example is the work of Marjorie Rice on pentagonal tilings of the plane.

Wikipeidia:

In December 1975, Rice came across a Scientific American article on tessellations. Despite having only a high-school education, she began devoting her free time to discovering new pentagonal tilings, ways to tile the plane using pentagons. She developed her own system of notation to represent the constraints on and relationships between the sides and angles of the polygons and used it to discover four new types of tessellating pentagons and over sixty distinct tessellations by pentagons by 1977. Rice's work was eventually examined by mathematics professor Doris Schattschneider, who deciphered the unusual notation and formally announced her discoveries to the mathematics community. Schattschneider has lauded Rice's work as an exciting discovery by an amateur mathematician.

That Wikipedia page links to a List of amateur mathematicians

Answer 3

There was a peculiar case when an anonymous forum commenter made significant progress on a problem. They could well have had substantial formal training (reading their posts I think this is likely).

The problem is establishing the formula for sequence A180632 of the On-Line Encyclopedia of Integer Sequences (OEIS): the Minimum length of a string of letters that contains every permutation of n letters as sub-strings, also known as length of the minimal super-permutation.

[In September 2011] one anonymous 4chan poster set out to solve this puzzle. The person claimed to have worked out a lower bound for the answer...

Crucially, they offered a proof – and, in doing so, they provided the world with an answer to a problem that had eluded mathematicians for a quarter of a century.

https://www.iflscience.com/editors-blog/an-anonymous-online-anime-fan-just-solved-a-problem-thats-been-eluding-mathematicians-for-decades/ (30 Oct 2018)

...because in 1993 there had been a publication on the problem that later turned out to be wrong. In October 2018, three professionals checked the proof and published a manuscript with Anonymous 4chan Poster as the first author:

The above mentioned lower bound was essentially shown in 2011 by an anonymous poster on the internet and, filling in some minor details, brought into a formal form by Houston, Pantone and Vatter (see reference).

http://oeis.org/A180632 (27 Oct 2018)