Permutations

• [Tutorial] A comprehensive guide to permutations for beginners by nor

Modular Arithmetic

• Modular Arithmetic for Beginners by Spheniscine
• If mod(%) are so expensive why not make own modulus? by Qualified

Binary Exponentiation

• Binary Exponentiation by cp-algo
• Calculating Series Sums with Binary Exponentiation by UnexpectedValue
• Binary Exponentiation by Errichto

Problems -- Set 1

Factors

• [tutorial]Counting Divisors of a Number in O(n^1/3) by himanshujaju

Euclidean algorithm

• Euclidean algorithm for computing the greatest common divisor by cp-algo

Extended Euclidean Algorithm

• Extended Euclidean Algorithm by cp-algo
• Extended Euclidean algorithm for n variables by Ahmed-Yasser
• Recovering a linear recurrence with the extended Euclidean algorithm by adamant

Linear Diophantine Equation

• Linear Diophantine Equation by cp-algo

Number of divisors / sum of divisors Function

• Number of divisors / sum of divisors by cp-algo

Modular Multiplicative Inverse

• Modular Multiplicative Inverse by cp-algo

Linear Congruence Equation

• Linear Congruence Equation by cp-algo

Euler's phi/totient function

• Euler's totient function by cp-algo
• [Tutorial] Euler's phi function, its properties, and how to compute it by kamilszymczak1

Sieve of Eratosthenes

• [Video Tutorial] Sieve of Eratosthenes by arujbansal
• Extensions of the Prime Sieve by Hikari9

Linear Sieve of Eratosthenes

• [Tutorial] Math note — linear sieve by Nisiyama_Suzune
• [Video Tutorial]Number Theoretic Functions and Linear Sieve by demoralizer

Multiplicative function

• Multiplicative function by Wikipedia
• [Tutorial] Math note — linear sieve by Nisiyama_Suzune
• Multiplicative Functions by BERKELEY MATH CIRCLE

Dirichlet convolution

• Dirichlet convolution by Wikipedia
• [Tutorial] Math note — Dirichlet convolution by Nisiyama_Suzune
• Dirichlet convolution and inverse by adamant
• Dirichlet convolution. Part 1: Fast prefix sum computations by adamant
• Dirichlet convolution. Part 2: Dirichlet series and prime counting by adamant
• Dirichlet series: generating functions for Dirichlet convolution by adamant

Möbius inversion.

• [Tutorial] Math note — Möbius inversion by Nisiyama_Suzune
• A Dance with Mobius Function by adurysk
• [Video Tutorial Unacademy] Number Theory (Dirichlet Convolution And Mobius Inversion) by ista2000
• [Video Tutorial]Advanced Number Theory: Mobius Inversion by demoralizer

Divisor function

• Divisor function by Wikipedia

Inclusion exclusion principle

Game theory

• Sprague-Grundy theorem. Nim by Cp-algo
• The Intuition Behind NIM and Grundy Numbers in Combinatorial Game Theory by Shisuko
• Advanced Game Theory by Kshitij Sodani by kshitij_sodani
• Introduction to Game Theory || Indian Programming Camp 2020 — Intermediate Track by adurysk
• Mastering CodeForces Game Theory Problems by YashDwivedi
• [Unacademy Official Class] Basics and Introduction To Game Theory In CP by adurysk
• Probability and Game Theory from CSES by acraider

Chinese Remainder Theorem

• Chinese Remainder Theorem by Cp-algo

Matrix Exponentiation

• [Tutorial]A Complete Guide on Matrix Exponentiation by lazyneuron
• Matrix Exponentiation tutorial + training contest by Errichto
• Cool tricks using Matrix Exponential by tweety
• Matrix Exponentiation Optimization by art-hack
• Nice trick involving sparse matrix exponentiation (kind-of) by bicsi
• Matrix exponentiation by ItsLastDay
• Matrix Exponentiation by Usaco.guide
• Matrix Exponentiation Practice Problems Mashup | Live stream with codes and explanationby demoralizer
• [Unacademy Official Class] Matrix Exponentiationby EnEm
• SnackDown 2021 Prep Series | Advanced Level | Matrix Exponentiationby EnEm
• Matrix Exponentiation and Matrix Operationsby demoralizer

Problems https://mirror.codeforces.com/gym/102644 , https://mirror.codeforces.com/gym/316783 , https://www.codechef.com/CCOD2020/problems/ECODOWN , https://www.codechef.com/problems/HXR

Catalan Numbers

• [Tutorial] Catalan Numbers and Catalan Convolution by Dardy

Factorial number system

• Factorial number system by Wikipedia

Generating Functions

• [Tutorial] Generating Functions in Competitive Programming (Part 1) by zscoder
• [Tutorial] Generating Functions in Competitive Programming (Part 2) by zscoder
• Combinatorial species: An intuition behind generating functions by adamant
• Useful substitutions with generating functions by adamant
• [Educational] Combinatorics Study Notes (5-1) | Generating Function For Beginners by Black_Fate
• Solving the "simple math problem" with generating functions by adamant

Group theory

Burnside's lemma

• Burnside's lemma / Pólya enumeration theorem by Cp-algo
• Orbit counting theorem or Burnside’s Lemma by GFG
• Burnside's Lemma by IMO Maths
• [Tutorial] Burnside's lemma (with example) by TwoFx
• Burnside Lemma made simple. by Everule
• [Video Tutorial] Burnside Lemma by acraider

Problems -- Set 1

Probability

• Sums and Expected Value — part 1 by Errichto
• Sums and Expected Value — part 2 by Errichto

FFT/NTT

• Tutorial on FFT/NTT — The tough made simple. ( Part 1 ) by sidhant
• Tutorial on FFT/NTT — The tough made simple. ( Part 2 ) by sidhant
• [Tutorial] FFT by -is-this-fft-
• CDQ convolution (online FFT) generalization with Newton method by adamant
• A simple sqrt decomposition solution to online FFT by jtnydv25
• FFT And Variants || Indian Programming Camp 2020 — Advanced Track by EnEm
• FFT and Convulutions by EnEm
• Divide & Conquer: FFT by Erik Demaine
• "Algorithms in Depth" Stream Series: Kicking Off With FFT by peltorator
• Algorithms in Depth: FFT Intermediate Stream by peltorator

• gcd(p,q)=gcd(p,p-q)
• Two integers a,b (b != 0) there exist two unique integers q and r such that a = bq + r, 0 <= r < b
• Bezout's identity states that if d = gcd(a,b) then there always exist integers x and y such that ax+by = d.
• number of primes smaller than n are n/log2(n)
• number of Divisors of a number n at max (n)^(1/3)
• number of prime Divisors of n at max log(n)
• harmonic series N + N/2 + N/3 + ... + 1 = O(N log N)
• (1+1/4 + 1/9 + 1/16 + 1/25 + ...) = (π^2 / 6)
• nCr = (n-1)C(r-1) + (n-1)Cr
• The total number of pairs (x,y) with 1≤x,y≤n,gcd(x,y)=1 is φ(1)+φ(2)+⋯+φ(n) , where φ is Euler's totient function.
• The number of divisors of a number n is (1+p1)*(1+p2)... where p1,p2..are the power of the prime factor of n that is n=a1^p1*a2^p2...
• N + N / 2 + N / 4 + N / 8 + N / 16 + ... is 2*N so it's O(N)
• Distinct values of the floor of (N / i) is 2 * sqrt(N) for i 1 to N.Proof
• There exists a k that n<=2^k<=2*n. Use in case we want to extend an array in the power of 2 still the Tc will be O(n)
• 1/1 +1/2 +1/3 ... 1/i.. =log(n) . i from 1 to n where n tends to inf
• set of bitwise AND of all subarray of an array such what ai<=n contain at max log(n) elements.
• A^B=C then A^C=B

More Statements are highly welcome.