Proofs in Mathematical Practice.- Motives for Finding Alternative Proofs.- Sums of Integers.- Quadratic Surds.- The Pythagorean Theorem.- The Fundamental Theorem of Arithmetic.- The Infinitude of the Primes.- The Fundamental Theorem of Algebra.- Desargues's Theorem.- The Prime Number Theorem.- The Irreducibility of the Cyclotomic Polynomials.- The Compactness of First-Order Languages.- Other Case Studies.