Download Special Facts to Know

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)