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Maybe I've missed it, but it seems no one has mentioned Pascal's triangle. According to Wikipedia it had a slew of earlier discoverers, going back to "The Persian mathematician Al-Karaji (953–1029) [w …

In 1917, Schur showed that there was no hope of proving Fermat's Last Theorem by ruling out the corresponding congruences. He applied a result from Ramsey Theory, but of course Ramsey Theory had not y …

Quoting https://en.wikipedia.org/wiki/Ax%E2%80%93Kochen_theorem, "The Ax–Kochen theorem, named for James Ax and Simon B. Kochen, states that for each positive integer $d$ there is a finite set $Y_d$ o …

The nonzero characteristics of fields are precisely the prime numbers.

The abelian simple groups are precisely the groups of prime order.

Eric Temple Bell, The Development of Mathematics, page 76: "Like Euclid in his explicit statement of the parallel postulate, Archimedes had the true mathematician's caution in the presence of the obvi …

There are stories about work of Banach being written up by people other than Banach. Details and debunking links available on another MO thread, Who wrote up Banach's Thesis?

Albert Gloden's book, Mehrgradige Gleichungen, was published in Groningen in 1944. Some of it is out of date, but it's still a good place to start the study of multigrade equations (equations in integ …

Conway's office at Cambridge was notoriously messy. One day, he got tired of how hard he had to struggle to find a paper in there, and shut himself away for a few hours to come up with a solution to t …

John McCleary will give a talk at the JMM in a couple of weeks on "Hassler Whitney and Fire Control in WWII." Whitney "was assigned to work on fire control, the mathematics of aiming weapons for accur …

Littlewood is best-known for his pure math research, but during the first World War he worked on ballistics. I suspect there were others who put aside pure math for more applied topics during the wars …

Logic and set theory were developed by Frege, Russell and Whitehead, Hilbert and others in the late 19th, early 20th centuries with the goal of providing a firm foundation for all of Mathematics. In t …

I quote at length from the Wikipedia essay on the history of knot theory: In 1867 after observing Scottish physicist Peter Tait's experiments involving smoke rings, Thomson came to the idea that …

The "class number one" problem for imaginary quadratic fields was posed by Gauss in 1798, settled independently by Baker and Stark in 1966/7. Gauss had found nine imaginary quadratic fields with class …

The Gelfond-Schneider Theorem, if $a$ and $b$ are algebraic numbers with $a\ne0,1$, and $b$ irrational, then any value of $a^b$ is a transcendental number, was proved independently in 1934 by Aleksand …