\contentsline {part}{I\hspace {1em}Fundamentals}{1}{part.1}%
\contentsline {chapter}{\numberline {1}Numbers}{3}{chapter.1}%
\contentsline {section}{\numberline {1.1}Types of numbers}{3}{section.1.1}%
\contentsline {subsection}{\numberline {1.1.1}Natural numbers}{3}{subsection.1.1.1}%
\contentsline {subsection}{\numberline {1.1.2}Integers}{4}{subsection.1.1.2}%
\contentsline {subsection}{\numberline {1.1.3}Rational numbers}{4}{subsection.1.1.3}%
\contentsline {section}{\numberline {1.2}Divisibility}{5}{section.1.2}%
\contentsline {subsection}{\numberline {1.2.1}Divisibility}{5}{subsection.1.2.1}%
\contentsline {subsection}{\numberline {1.2.2}Euclid's division lemma}{5}{subsection.1.2.2}%
\contentsline {subsection}{\numberline {1.2.3}Multiples and factors}{6}{subsection.1.2.3}%
\contentsline {subsection}{\numberline {1.2.4}Lowest common multiple}{6}{subsection.1.2.4}%
\contentsline {subsection}{\numberline {1.2.5}Greatest common factor}{6}{subsection.1.2.5}%
\contentsline {subsection}{\numberline {1.2.6}Euclidean algorithm}{7}{subsection.1.2.6}%
\contentsline {subsection}{\numberline {1.2.7}Bézout's lemma}{7}{subsection.1.2.7}%
\contentsline {section}{\numberline {1.3}Primality}{8}{section.1.3}%
\contentsline {subsection}{\numberline {1.3.1}Prime numbers}{8}{subsection.1.3.1}%
\contentsline {subsection}{\numberline {1.3.2}Euclid's theorem}{9}{subsection.1.3.2}%
\contentsline {subsection}{\numberline {1.3.3}Pair of coprime numbers}{9}{subsection.1.3.3}%
\contentsline {subsection}{\numberline {1.3.4}Euclid's lemma}{10}{subsection.1.3.4}%
\contentsline {subsection}{\numberline {1.3.5}Naive factorization algorithm}{10}{subsection.1.3.5}%
\contentsline {subsection}{\numberline {1.3.6}Fundamental theorem of arithmetic}{11}{subsection.1.3.6}%
\contentsline {subsection}{\numberline {1.3.7}Types of prime numbers}{11}{subsection.1.3.7}%
\contentsline {chapter}{\numberline {2}Modular arithmetic}{13}{chapter.2}%
\contentsline {section}{\numberline {2.1}Introduction to modular arithmetic}{13}{section.2.1}%
\contentsline {subsection}{\numberline {2.1.1}Basic laws}{13}{subsection.2.1.1}%
\contentsline {section}{\numberline {2.2}Divisibility tests}{15}{section.2.2}%
\contentsline {subsection}{\numberline {2.2.1}Modular multiplicative inverse}{15}{subsection.2.2.1}%
\contentsline {section}{\numberline {2.3}Euler's theorem}{15}{section.2.3}%
\contentsline {subsection}{\numberline {2.3.1}Fermat's little theorem}{16}{subsection.2.3.1}%
\contentsline {subsection}{\numberline {2.3.2}Euler's totient function}{16}{subsection.2.3.2}%
\contentsline {subsection}{\numberline {2.3.3}Euler's theorem}{16}{subsection.2.3.3}%
\contentsline {subsection}{\numberline {2.3.4}Primitive roots}{17}{subsection.2.3.4}%
\contentsline {subsection}{\numberline {2.3.5}Discrete logarithms}{17}{subsection.2.3.5}%
\contentsline {section}{\numberline {2.4}Chinese remainder theorem}{18}{section.2.4}%
\contentsline {section}{\numberline {2.5}Quadratic residues}{19}{section.2.5}%
\contentsline {subsection}{\numberline {2.5.1}Modular quadratics}{19}{subsection.2.5.1}%
\contentsline {subsection}{\numberline {2.5.2}Lagrange's theorem}{20}{subsection.2.5.2}%
\contentsline {subsection}{\numberline {2.5.3}Law of quadratic reciprocity}{22}{subsection.2.5.3}%
\contentsline {section}{\numberline {2.6}Ring theoretic formulation}{22}{section.2.6}%
\contentsline {subsection}{\numberline {2.6.1}Additive group of integers modulo $n$}{23}{subsection.2.6.1}%
\contentsline {subsection}{\numberline {2.6.2}Multiplicative group of integers modulo $n$}{23}{subsection.2.6.2}%
\contentsline {subsection}{\numberline {2.6.3}Commutative ring of integers modulo $n$}{23}{subsection.2.6.3}%
\contentsline {chapter}{\numberline {3}Integer sequences}{25}{chapter.3}%
\contentsline {section}{\numberline {3.1}Introduction to numeric sequences}{25}{section.3.1}%
\contentsline {section}{\numberline {3.2}Parity}{26}{section.3.2}%
\contentsline {section}{\numberline {3.3}Arithmetic progression}{26}{section.3.3}%
\contentsline {section}{\numberline {3.4}Geometric progression}{26}{section.3.4}%
\contentsline {section}{\numberline {3.5}$n$-gonal numbers}{27}{section.3.5}%
\contentsline {section}{\numberline {3.6}Recursive sequences}{27}{section.3.6}%
\contentsline {subsection}{\numberline {3.6.1}Fibonacci sequence}{27}{subsection.3.6.1}%
\contentsline {subsection}{\numberline {3.6.2}Lucas sequence}{28}{subsection.3.6.2}%
\contentsline {subsection}{\numberline {3.6.3}Pell sequence}{28}{subsection.3.6.3}%
\contentsline {chapter}{\numberline {4}Rational sequences}{29}{chapter.4}%
\contentsline {section}{\numberline {4.1}Bernoulli numbers}{29}{section.4.1}%
\contentsline {subsection}{\numberline {4.1.1}Sums of powers}{29}{subsection.4.1.1}%
\contentsline {subsection}{\numberline {4.1.2}Bernoulli numbers}{30}{subsection.4.1.2}%
\contentsline {subsection}{\numberline {4.1.3}Properties of Bernoulli numbers}{31}{subsection.4.1.3}%
\contentsline {subsection}{\numberline {4.1.4}Applications of Bernoulli numbers}{31}{subsection.4.1.4}%
\contentsline {section}{\numberline {4.2}Harmonic numbers}{31}{section.4.2}%
\contentsline {chapter}{\numberline {5}Elementary approach to Diophantine equations}{33}{chapter.5}%
\contentsline {section}{\numberline {5.1}Linear Diophantine equations}{33}{section.5.1}%
\contentsline {subsection}{\numberline {5.1.1}Linear Diophantine equation}{34}{subsection.5.1.1}%
\contentsline {subsection}{\numberline {5.1.2}Geometric analysis}{34}{subsection.5.1.2}%
\contentsline {section}{\numberline {5.2}Homogeneous Diophantine equations}{35}{section.5.2}%
\contentsline {subsection}{\numberline {5.2.1}Pythagorean triples}{35}{subsection.5.2.1}%
\contentsline {subsection}{\numberline {5.2.2}Sum of two squares}{35}{subsection.5.2.2}%
\contentsline {chapter}{\numberline {6}Elementary arithmetic functions}{37}{chapter.6}%
\contentsline {section}{\numberline {6.1}Arithmetic function}{37}{section.6.1}%
\contentsline {subsection}{\numberline {6.1.1}Arithmetic function}{37}{subsection.6.1.1}%
\contentsline {subsection}{\numberline {6.1.2}Additivity and multiplicativity}{37}{subsection.6.1.2}%
\contentsline {subsection}{\numberline {6.1.3}Examples of familiar arithmetic functions}{38}{subsection.6.1.3}%
\contentsline {section}{\numberline {6.2}Divisor functions}{38}{section.6.2}%
\contentsline {subsection}{\numberline {6.2.1}Divisor functions}{38}{subsection.6.2.1}%
\contentsline {subsection}{\numberline {6.2.2}Tau function}{38}{subsection.6.2.2}%
\contentsline {subsection}{\numberline {6.2.3}Higher order divisor function}{39}{subsection.6.2.3}%
\contentsline {subsection}{\numberline {6.2.4}Perfect numbers}{39}{subsection.6.2.4}%
\contentsline {section}{\numberline {6.3}Totient functions}{39}{section.6.3}%
\contentsline {subsection}{\numberline {6.3.1}Euler's totient function}{39}{subsection.6.3.1}%
\contentsline {subsection}{\numberline {6.3.2}Jordan's totient function}{40}{subsection.6.3.2}%
\contentsline {subsection}{\numberline {6.3.3}Carmichael function}{40}{subsection.6.3.3}%
\contentsline {section}{\numberline {6.4}Multiplicative function}{41}{section.6.4}%
\contentsline {subsection}{\numberline {6.4.1}Möbius function}{41}{subsection.6.4.1}%
\contentsline {subsection}{\numberline {6.4.2}Liouville function}{41}{subsection.6.4.2}%
\contentsline {subsection}{\numberline {6.4.3}Partition function}{42}{subsection.6.4.3}%
\contentsline {subsection}{\numberline {6.4.4}Von Mangoldt function}{42}{subsection.6.4.4}%
\contentsline {part}{II\hspace {1em}Advanced}{43}{part.2}%
\contentsline {chapter}{\numberline {7}$p$-adic numbers}{45}{chapter.7}%