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Landau's four problems are now over a century old (1912), and each still unsolved. This seems remarkable, even though he was not the originating author all four (maybe only the 4th?). Still, he isolat …

Great Cubicuboctahedron Great Icosacronic Hexecontahedron Great Rhombic Triacontahedron Great Snub Icosidodecahedron Great Stellated Dodecahedron Great Triakis Octahedron ... There are many polyhedr …

Tristan Needham says (p.174),* "While Gauss and Bonnet certainly paved the road to [the Gauss-Bonnet Theorem], neither one of them was even aware of this extraordinary result, let alone stated it!" …

John Horton Conway is known for many achievements: Life, the three sporadic groups in the "Conway constellation," surreal numbers, his "Look-and-Say" sequence analysis, the Conway-Schneeberger $15$-th …

It may be that certain theorems, when proved true, counterintuitively retard progress in certain domains. Lloyd Trefethen provides two examples: Faber's Theorem on polynomial interpolation: Interpre …

The title is a quote from a Jim Holt article entitled, "The Riemann zeta conjecture and the laughter of the primes" (p. 47).1 His example of a "long-standing conjecture" is the Riemann hypothesis, and …

From my limited perspective, it appears that the understanding of a mathematical phenomenon has usually been achieved, historically, in a continuous setting before it was fully explored in a discrete …

Yitang Zhang's Annals of Mathematics primes-gap result opened a new window, which Polymath's reduction from $70\times 10^6$ to $246$ attests. Perhaps Harald Helfgott's celebrated proof of the odd Gol …

Q. When did the notion of homeomorphism reach its modern formulation as a bicontinuous bijection, i.e., a continuous bijection between topological spaces whose inverse is also continuous? …

The ancient Babylonians understood squares:       Plimpton 322 The ancient Athenians understood cubes, if we can take doubling the cube, i.e., the Delian problem, as evidence. My question is: Q. …

Archimedes (ca. 287-212BC) described what are now known as the 13 Archimedean solids in a lost work, later mentioned by Pappus. But it awaited Kepler (1619) for the 13 semiregular polyhedra to be rec …

Expander graphs ("sparse graphs that have strong connectivity properties") burst onto the mathematical scene around the millennium, but I have not been successful in tracing the origin of (a) the conc …

"Adleman refers to integers which factor completely into small primes as “smooth” numbers." (ME Hellman, JM Reyneri. Advances in Cryptology, 1983: citation link.) Does anyone know what is the int …

I have hit upon major (for me—relative to my trivial accomplishments) insights in my research in various sleep-deprived altered states of consciousness, e.g., long solo car-drives extending through th …

Thales (circa 600BC—roughly 50 years before Pythagoras, 200 years before Plato, and 300 years before Euclid) certainly knew and reasoned with the concept of a planar angle. Are there earlier historica …