What is John Charles Martin Nash known for? [closed]

Question

John Nash and his wife Alicia tragically passed away in 2015. According to Sylvia Nasar's book "A Beautiful Mind", their son is apparently a good mathematician. What works is he known for?

Answer 1

tl;dr — a handful of papers on number theory, particularly of the Paul Erdős variety.

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To be clear, we are talking about John Charles Martin Nash, who was born in 1959, and got his PhD in 1985 from Rutgers, not John Forbes Nash Jr.

Google returns an article from 2017 with this quote:

"The PhD in mathematics from Rutgers University said he "passes the time" playing chess and math games online with opponents around the world. He pores over his monthly chess magazine, and keeps up with news on the internet and television."

Here's the list of publications attributed to him on zbMATH. The data quality is not great, with lots of false positives! The paper

• John C. Nash, A one-sided transformation method for the singular value decomposition and algebraic eigenproblem. The Computer Journal, Volume 18, Issue 1, 1975, Pages 74–76 https://doi.org/10.1093/comjnl/18.1.74

would have been when he was 16, but a work address is given, at 'Agriculture Canada'. So clearly not the same guy. Similarly, other papers/books on computer-related work are not by him, for example

• J.C. Nash, Minimizing a Non-linear Sum of Squares Function on a Small Computer, IMA Journal of Applied Mathematics, 19(2) (1997) 231–237. https://doi.org/10.1093/imamat/19.2.231

or the 1979 book Compact Numerical Methods for Computers, or the the 2014 book Nonlinear Parameter Optimization Using R Tools.

And the John C. Nash who published

• John C. Nash, Positive Ricci curvature on fibre bundles. J. Differential Geom. 14(2): 241-254 (1979). https://doi.org/10.4310/jdg/1214434973

is not him either. He would have been 20, but working at the University of Colorado, Colorado Spring. That author seems to be this John C. Nash.

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So what about the actual works by the man in question? This is co-written with his PhD advisor on a relevant topic:

• Nash, John C. M.; Nathanson, Melvyn B., Cofinite subsets of asymptotic bases for the positive integers, Journal of Number Theory 20 Issue 3 (1985) Pages 363-372, https://doi.org/10.1016/0022-314X(85)90027-7

And this one lists a Rutgers address:

• John C.M. Nash, Freiman's theorem answers a question of Erdös, Journal of Number Theory 27 Issue 1 (1987) Pages 7-8, https://doi.org/10.1016/0022-314X(87)90044-8

The paper

• John C. M. Nash, On $B_4$-Sequences, Canadian Mathematical Bulletin, 32 Issue 4 (1989) pp. 446-449, https://doi.org/10.4153/CMB-1989-064-2

then lists an address of Department of Mathematics, Marshall University, in West Virginia (note that this is the state where J.C.M. Nash's father was born). The topic of this paper is close to the previous paper. Nash spoke about this work at an AMS sectional meeting in April 1989.

We then get

• John C.M. Nash, Some Applications of a Theorem of M. Kneser, Journal of Number Theory 44 Issue 1 (1993) Pages 1-8, https://doi.org/10.1006/jnth.1993.1027

now with a Rutgers address again. And lastly we get a very short note publishing a minor observation:

• John C. M. Nash, Hyperperfect numbers, Period. Math. Hung. 45 No. 1-2 (2002) pp 121-122, https://doi.org/10.1023/A:1022306315474

now with a Princeton affiliation. You can see J.C.M. Nash wearing a Princeton t-shirt in the photos in the article, for what it's worth. Also, John his father and Alicia were dedicated to looking after him, and John F Nash was working at Princeton.

So, in summary: $4+\varepsilon$ papers on number theory, and he seemed to have had positions at Rutgers U, Marshall U and at least an affiliation at Princeton, but I can't find independent evidence for the latter two.