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Special Numbers Transcendental – not a root of a polynomial with integer coefficients Unit – has a multiplicative inverse Fibonacci – F0 = 0, F1 = 1, Fn = Fn-1 + Fn-2 for n > 1 Lucas – F0 = 2, F1 = 1, Fn = Fn-1 + Fn-2 for n > 1 Let s(n) be the sum of all the proper factors of n. Deficient – s(n) < n Perfect – s(n) = n Abundant – s(n) > n Let d(n) be the total number of digits in the prime factorization of n. Frugal / Economical – d(n) < n Equidigital – d(n) = n Extravagant / Wasteful – d(n) > n Let b(n) be the number of 1s in the binary representation of n. Evil – b(n) is even Odious – b(n) is odd Happy – summing the squares of the digits of n eventually results in 1 Lucky – remain when every kth number is removed from a list of positive integers, where k enumerates over all lucky numbers greater than 1 Polite – not a power of 2 Primeval – smallest integer where k permutations of its digits are prime, for some k 1, 2, 13, 37, 107, 113, 137… Other (more obvious) numbers: Odd and even numbers Prime and composite numbers Natural numbers 1, 2, 3… Whole numbers 0, 1, 2, 3… Integers …-3, -2, -1, 0, 1, 2, 3… Rational and irrational numbers Real, imaginary, and complex numbers Mathematicians Euclid – wrote Elements with rigorous proofs in geometry and number theory Archimedes – used infinitesimals to calculate areas / volumes of curves; Archimedean spiral Eratosthenes – thought Earth was round, computed radius of Earth; Sieve of Eratosthenes Claudius Ptolemy – wrote Algamest; made geocentric system Diophantus – wrote Arithmetica; father of algebra; Diophantine equations Hypatia – considered first notable female mathematician; invented hydrometer Theano – wife of Pythagoras; wrote treatise on golden mean Rene Descartes – invented analytic geometry; Cartesian plane; Descartes’s rule of signs John Napier – invented logarithms; Napier’s bones Leibniz – invented infinitesimal calculus Euler – everything; contributions to calculus, number theory, graph theory, and notation Goldbach – Goldbach’s conjecture; perfect-powers theorem in number theory Agnesi – wrote first book on differential & integral calculus; Witch of Agnesi curve Sophie Germain – contributions to Fermat’s Last Theorem and elasticity Boole – founder of Boolean logic & Boolean algebra Lady Lovelace (Ada Byron) – 1st programmer; wrote algorithm to compute Bernoulli numbers on Babbage’s machine John Venn – Venn diagram; contributions to logic & probability Georg Cantor – invented set theory Sonya Kovalevsky – contributions to analysis; Kovalevsky top Alicia Scott – studied 4D polytopes Emmy Noether – studied abstract algebra; taught “Noether’s boys” Grace Alele Williams – first Nigerian woman with doctorate; studied mathematical education Karen Smith – studied commutative algebra & algebraic geometry Freda Porter – first Native American woman with doctorate; studied applied mathematics with computer science Mandelbrot – studied fractals, finance, and fluid dynamics Lucky Numbers Program (simple) Input N seq(2X-1, X, 1, N/2) → L1 2 → X While L1(X) < dim(L1) 0 → Y For (I, 1, dim(L1)) If fPart(I / L1(X)) Then Y + 1 → Y L1(I) → L1(Y) End End Y → dim(L1) X + 1 → X End (fast) Input N seq(2X-1, X, 1, N/2) → L1 2 → X dim(L1) → D While L1(X) < D D – int(D / L1(X)) → D For (I, 1, D) L1(I + int((I-1) / (L1(X)-1))) → L1(I) End X + 1 → X End D → dim(L1)