Mathematical algorithms use mathematical concepts, formulas, and properties to solve computational problems efficiently. They are commonly used in problems involving numbers, arithmetic, number theory, algebra, and sequences.
- They include algorithms for problems such as GCD, LCM, divisibility, prime numbers, modular arithmetic, and number systems.
- They help apply mathematical properties to develop efficient solutions for programming and problem-solving.
GCD and LCM
- GCD of Two Numbers
- LCM of Two Numbers
- LCM of array
- GCD of array
- Basic and Extended Euclidean
- Stein’s Algorithm for GCD
- GCD, LCM and Distributive Property
- Count pairs with gcd (A, B) as B
- GCD of floating point numbers
- Series with largest GCD and sum equals to n
- Largest Subset with GCD 1
- Summation of GCD of all the pairs up to N
Divisibility & Large Numbers
Theory: Mathematical Algorithms - Divisibility and Large Numbers
- Divisibility by 3
- Divisibility by 4
- Divisibility by 6
- Divisibility by 7
- Divisibility by 9
- Divisibility by 11
- Divisibility by 12
- Divisibility by 13
- Divisibility by 15
- Divisibility by 29
Series
Theory: Series
- Juggler Sequence
- Padovan Sequence
- Aliquot Sequence
- Moser-de Bruijn Sequence
- Stern-Brocot Sequence
- Newman-Conway Sequence
- Sylvester’s sequence
- Recaman’s sequence
- Sum of 2, 22, 222, ………
- Sum of 1^2 + 3^2 + 5^2 + . . . + (2*n – 1)^2
- Sum of 0.6, 0.06, 0.006, 0.0006, …to n terms
- n-th term of 2, 12, 36, 80, 150….
Number Digits
- Minimum digits to remove to make a number Perfect Square
- Print first k digits of 1/n where n is a positive integer
- Check if a given number can be represented in given a no. of digits in any base
- Find element using minimum segments in Seven Segment Display
- Find next greater number with same set of digits
- Check if a number is jumbled or not
- Numbers having difference with digit sum more than s
- Total numbers with no repeated digits in a range
- K-th digit in ‘a’ raised to power ‘b’
Algebra
Theory: Algebra
- Program to add two polynomials
- Multiply two polynomials
- Find number of solutions of a linear equation of n variables
- Calculate the Discriminant Value
- Program for dot product and cross product of two vectors
- Iterated Logarithm log*(n)
- Program to find correlation coefficient
- Program for Muller Method
- Number of non-negative integral solutions of a + b + c = n
- Generate Pythagorean Triplets
Number System
Theory: Number System
- Exponential notation of a decimal number
- Check if a number is power of k using base changing method
- Convert a binary number to hexadecimal number
- Program for decimal to hexadecimal conversion
- Converting a Real Number (between 0 and 1) to Binary String
- Convert from any base to decimal and vice versa
- Decimal to binary conversion without using arithmetic operators
Prime Numbers & Primality Tests
Theory: Prime numbers and Primality Tests
- Prime Numbers
- Left-Truncatable Prime
- Mersenne Prime
- Super Prime
- Hardy-Ramanujan Theorem
- Rosser’s Theorem
- Fermat’s little theorem
- Program for Prime Number Check
- Vantieghems Theorem for Primality Test
- AKS Primality Test
- Lucas Primality Test
Prime Factorization & Divisors
Theory: Prime Factorization and Divisors
- Prime factors
- Smith Numbers
- Sphenic Number
- Hoax Number
- k-th prime factor of a given number
- Pollard’s Rho Algorithm for Prime Factorization
- Finding power of prime number p in n!
- Find all divisors of a natural number
- Find numbers with n-divisors in a given range
Modular Arithmetic
Theory: Modular Arithmetic
- Modular Exponentiation (Power in Modular Arithmetic)
- Modular multiplicative inverse
- Modular Division
- Euler’s criterion (Check if square root under modulo p exists)
- Find sum of modulo K of first N natural number
- How to compute mod of a big number?
- Exponential Squaring (Fast Modulo Multiplication)
- Trick for modular division ( (x1 * x2 …. xn) / b ) mod (m)
Factorial
Theory: Factorial
- Program for factorial of a number
- Legendre’s formula (Given p and n, find the largest x such that p^x divides n!)
- Count trailing zeroes in factorial of a number
- Factorial of a large number
- Primorial of a number
- Find maximum power of a number that divides a factorial
- Largest power of k in n! (factorial) where k may not be prime
- Check if a number is a Krishnamurthy Number or not
- Last non-zero digit of a factorial
- Count digits in a factorial using Logarithm
Fibonacci Numbers
Theory: Fibonacci Sequence
- Fibonacci Numbers
- Interesting facts about Fibonacci numbers
- Zeckendorf’s Theorem (Non-Neighbouring Fibonacci Representation)
- Finding nth Fibonacci Number using Golden Ratio
- Matrix Exponentiation
- Fibonacci Coding
- Cassini’s Identity
- Tail Recursion for Fibonacci
Catalan Numbers
Theory: Catalan Numbers
- Catalan numbers
- Applications of Catalan Numbers
- Dyck path
- Succinct Encoding of Binary Tree
- Find the number of valid parentheses expressions of given length
nCr Computations
Theory: nCr| CombinationsÂ
- Binomial Coefficient
- Introduction and Dynamic Programming solution to compute nCr%p
- Program to calculate value of nCr
- Rencontres Number (Counting partial derangements)
- Sum of squares of binomial coefficients
- Space and time efficient Binomial Coefficient
- Horner’s Method for Polynomial Evaluation
Set Theory
Theory: Set Theory
- Power Set
- Minimize the absolute difference of sum of two subsets
- Sum of all subsets of a set formed by first n natural numbers
- Sum of average of all subsets
- Bell Numbers (Number of ways to Partition a Set)
Sieve Algorithms
Theory: Sieve of Eratosthenes
- Sieve of Eratosthenes
- Segmented Sieve
- Sieve of Atkin
- Sieve of Sundaram to print all primes smaller than n
- Sieve of Eratosthenes in 0(n) time complexity
- Prime Factorization using Sieve O(log n) for multiple queries
Euler Totient Function
Theory: Euler TotientÂ
- Euler’s Totient Function
- Optimized Euler Totient Function for Multiple Evaluations
- Euler’s Totient function for all numbers smaller than or equal to n
- Primitive root of a prime number n modulo n
- Euler’s Four Square Identity
Chinese Remainder Theorem
Theory: Chinese Remainder Theorem
- Introduction to Chinese Remainder Theorem
- Implementation of Chinese Remainder theorem (Inverse Modulo based implementation)
- Cyclic Redundancy Check and Modulo-2 Division
- Using Chinese Remainder Theorem to Combine Modular equations
Some Practice Problems
- Interquartile Range (IQR)
- Simulated Annealing
- Pseudo Random Number Generator (PRNG)
- Square root of a number using log
- Find ways an Integer can be expressed as sum of n-th power of unique natural numbers
- N-th root of a number
- Fast Fourier Transformation for poynomial multiplication
- Find Harmonic mean using Arithmetic mean and Geometric mean
- Double Base Palindrome
- Program for Derivative of a Polynomial
- Sgn value of a polynomial
- Check if a number is a power of another number
- Program to evaluate simple expressions
- Make a fair coin from a biased coin
- Implement *, – and / operations using only + arithmetic operator
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