Mathematical Algorithms

The following is the list of mathematical coding problem ordered topic wise. Please refer Mathematical Algorithms (Difficulty Wise) for the difficulty wise list of problems.

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:

• 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:

• 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:

• 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:

• 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:

• Prime Numbers
• Left-Truncatable Prime
• Mersenne Prime
• Super Prime
• Hardy-Ramanujan Theorem
• Rosser’s Theorem
• Fermat’s little theorem
• Introduction to Primality Test and School Method
• Vantieghems Theorem for Primality Test
• AKS Primality Test
• Lucas Primality Test

Prime Factorization & 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:

• 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:

• 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:

• 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:

• 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:

• 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:

• 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:

• 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:

• 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:

• 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

Quick Links :

• Learn Data Structure and Algorithms | DSA Tutorial
• Practrice Problems on Mathematical Algorithms
• Quizzes on Mathematical Algorithms
• Videos on Mathematical Algorithms