Competencies: Mathematics > Number Theory

Number Theory

Body of Knowledge

Topic	Description	Relevance	Career Tracks
Divisibility	Division algorithm, divisors, multiples, divisibility rules, well-ordering principle as foundation for induction.	High	Cryptographer, Security Engineer, Software Engineer
Primes & Factorization	Fundamental theorem of arithmetic, prime testing, Sieve of Eratosthenes, prime factorization algorithms, infinitude of primes.	Critical	Cryptographer, Security Engineer, Research Mathematician
GCD, LCM & Euclidean Algorithm	Greatest common divisor, least common multiple, extended Euclidean algorithm, Bezout’s identity, applications to modular inverses.	Critical	Cryptographer, Security Engineer, Software Engineer
Modular Arithmetic	Congruences, residue classes, arithmetic in Z/nZ, linear congruences, modular exponentiation, applications to hashing and cryptography.	Critical	Cryptographer, Security Engineer, Software Engineer
Chinese Remainder Theorem	System of simultaneous congruences, constructive solution, applications to RSA optimization and distributed systems.	High	Cryptographer, Security Engineer
Fermat’s Little Theorem	a^(p-1) = 1 (mod p) for prime p, primality testing (Fermat test), pseudoprimes and Carmichael numbers.	Critical	Cryptographer, Security Engineer
Euler’s Theorem	a^(phi(n)) = 1 (mod n), Euler’s totient function, RSA encryption/decryption foundation, computing totients.	Critical	Cryptographer, Security Engineer
Quadratic Reciprocity	Legendre symbol, quadratic residues, law of quadratic reciprocity, Jacobi symbol extension.	Medium	Cryptographer, Research Mathematician
Diophantine Equations	Integer solutions to polynomial equations, linear Diophantine equations, Pythagorean triples, Pell’s equation.	Medium	Research Mathematician, Cryptographer
Continued Fractions	Simple continued fraction expansion, convergents, best rational approximations, applications to Diophantine approximation.	Medium	Research Mathematician, Cryptographer
Prime Distribution	Prime counting function, prime number theorem, Chebyshev bounds, Riemann zeta function connection, density of primes.	Medium	Research Mathematician, Cryptographer

Topic

Description

Relevance

Career Tracks

Divisibility

Division algorithm, divisors, multiples, divisibility rules, well-ordering principle as foundation for induction.

High

Cryptographer, Security Engineer, Software Engineer

Primes & Factorization

Fundamental theorem of arithmetic, prime testing, Sieve of Eratosthenes, prime factorization algorithms, infinitude of primes.

Critical

Cryptographer, Security Engineer, Research Mathematician

GCD, LCM & Euclidean Algorithm

Greatest common divisor, least common multiple, extended Euclidean algorithm, Bezout’s identity, applications to modular inverses.

Critical

Cryptographer, Security Engineer, Software Engineer

Modular Arithmetic

Congruences, residue classes, arithmetic in Z/nZ, linear congruences, modular exponentiation, applications to hashing and cryptography.

Critical

Cryptographer, Security Engineer, Software Engineer

Chinese Remainder Theorem

System of simultaneous congruences, constructive solution, applications to RSA optimization and distributed systems.

High

Cryptographer, Security Engineer

Fermat’s Little Theorem

a^(p-1) = 1 (mod p) for prime p, primality testing (Fermat test), pseudoprimes and Carmichael numbers.

Critical

Cryptographer, Security Engineer

Euler’s Theorem

a^(phi(n)) = 1 (mod n), Euler’s totient function, RSA encryption/decryption foundation, computing totients.

Critical

Cryptographer, Security Engineer

Quadratic Reciprocity

Legendre symbol, quadratic residues, law of quadratic reciprocity, Jacobi symbol extension.

Medium

Cryptographer, Research Mathematician

Diophantine Equations

Integer solutions to polynomial equations, linear Diophantine equations, Pythagorean triples, Pell’s equation.

Medium

Research Mathematician, Cryptographer

Continued Fractions

Simple continued fraction expansion, convergents, best rational approximations, applications to Diophantine approximation.

Medium

Research Mathematician, Cryptographer

Prime Distribution

Prime counting function, prime number theorem, Chebyshev bounds, Riemann zeta function connection, density of primes.

Medium

Research Mathematician, Cryptographer

Personal Status

To be populated after initial study and self-assessment.