Proofs from THE BOOK (eBook)

Number Theory.- Six proofs of the infinity of primes.- Bertrand’s postulate.- Binomial coefficients are (almost) never powers.- Representing numbers as sums of two squares.- The law of quadratic reciprocity.- Every finite division ring is a field.- Some irrational numbers.- Three times ?²/6.- Geometry.- Hilbert’s third problem: decomposing polyhedra.- Lines in the plane and decompositions of graphs.- The slope problem.- Three applications of Euler’s formula.- Cauchy’s rigidity theorem.- Touching simplices.- Every large point set has an obtuse angle.- Borsuk’s conjecture.- Analysis.- Sets, functions, and the continuum hypothesis.- In praise of inequalities.- The fundamental theorem of algebra.- One square and an odd number of triangles.- A theorem of Pólya on polynomials.- On a lemma of Littlewood and Offord.- Cotangent and the Herglotz trick.- Buffon’s needle problem.- Combinatorics.- Pigeon-hole and double counting.- Tiling rectangles.- Three famous theorems on finite sets.- Shuffling cards.- Lattice paths and determinants.- Cayley’s formula for the number of trees.- Identities versus bijections.- Completing Latin squares.- Graph Theory.- The Dinitz problem.- Five-coloring plane graphs.- How to guard a museum.- Turán’s graph theorem.- Communicating without errors.- The chromatic number of Kneser graphs.- Of friends and politicians.- Probability makes counting (sometimes) easy.