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A long-standing conjecture in Number Theory is that for each positive integer $n$ there is no stretch of $n$ consecutive integers containing more primes than the stretch from 2 to $n+1$. Just looking …

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 …

Just today, I read in the July 2014 Bulletin of the American Math Society, in the Mathematical Perspectives piece by Gerald Alexanderson, that "Lorenzo Mascheroni ... in ... 1797, proved that any [str …

This one happened - I was there (as an observer, not a principal). Only the names have been changed. X was Professor A's first doctoral student, and their relations weren't good. Rumor had it that t …

Bruce Berndt writes, Most of Ramanujan's mistakes arise from his claims in analytic number theory, where his unrigorous methods led him astray. In particular, Ramanujan thought his approximations an …

I have no idea whether this one is true - I heard it at Harvard, around 1970. The story goes that a PhD student was so sure no one would ever read his dissertation that he stuck in the middle of it an …

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 …

Siegel published The integer solutions of the equation $y^2=ax^n+bx^{n-1}+\cdots+k$, J London Math Soc 1 (1926) 66-68, under the pseudonym, X. Anecdotal evidence of a non-pseudonym: Once when Little …

A Group of Order 8,315,553,613,086,720,000 by J H Conway, Bull. London Math. Soc. (1969) 1 (1): 79-88, https://doi.org/10.1112/blms/1.1.79 Maybe it's cheating to call this memorable - I remembered the …

According to Gordan, the Hilbert Basis Theorem was an application of theology.

The original question, and several of the answers, refer to misuse of Godel's work, but with very few specific citations. For these, I would suggest Torkel Franzen's book, Godel's Theorem: An Incomple …

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 …

In the lengthy review by Victor J. Katz of The Oxford Handbook of the History of Mathematics, edited by Eleanor Robson and Jacqueline Stedall, Oxford University Press, Oxford, 2009, MR2549261 (2011e:0 …

Euclid's proofs were accepted for two thousand years. Only in the late 19th century was it noticed by Hilbert and others that Euclid was making a lot of implicit assumptions and that if you don't make …

You may want to look into the history of the de Branges proof of the Bieberbach conjecture. Reader's Digest version: his original proof was over 100 pages, but others studying his proof got it down to …