Mathematical Algorithms | Prime numbers and Primality Tests

What is Prime Number?

A prime number is a natural number greater than 1 that is divisible by only 1 and the number itself.

In other words, the prime number can be defined as a positive integer greater than that has exactly two factors, 1 and the number itself. First 10 prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.

Note: 1 is neither a prime number nor a composite number. All other numbers can be classified as prime or composite.

Some important theorems related to prime numbers are mentioned here:

• Hardy-Ramanujan Theorem
• Rosser’s Theorem
• Fermat’s little theorem
• Euclid Euler Theorem
• Euclid’s Lemma
• Bertrand’s Postulate
• Legendre’s Conjecture

What is Primality Test?

Primality Test is an algorithm to determine whether a number is prime or not.

Unlike integer factorization, primality test only states if the input is prime or not and does not generally provide prime factors. Some of the frequently used primality test algorithms are stated below

Some practice problems on Prime Numbers:

• Easy:
  1. Prime Numbers
  2. Left-Truncatable Prime
  3. Mersenne Prime
  4. Super Prime
  5. Palindromic Primes
  6. Almost Prime Numbers
  7. Pernicious number
  8. Twisted Prime Number
  9. Newman–Shanks–Williams prime
  10. Recursive program for prime number
  11. Co-prime pair with given sum minimum difference
  12. Check whether a number is Emirpimes or not
  13. Check if a number is sandwiched between primes
  14. Check whether a number is semiprime or not
  15. Check if two numbers are co-prime or not
• Intermediate:
  1. Special prime numbers
  2. Sophie Germain Prime
  3. Balanced Prime
  4. Prime Triplet
  5. 2^x + 1 and Prime
  6. Check if a number is Full Prime
  7. Check whether a number is circular prime or not
  8. Check whether given three numbers are adjacent primes
  9. Print prime numbers from 1 to N in reverse order
  10. Program to find sum of prime numbers between 1 to n
  11. Find two prime numbers with given sum
  12. Count Primes in Ranges
  13. Queries for maximum difference between prime numbers in given ranges
• Hard:
  1. Twin Prime Numbers
  2. Pierpont Prime
  3. Prime numbers and Fibonacci
  4. The Ulam Spiral
  5. Find coordinates of a prime number in a Prime Spiral
  6. Program for Goldbach’s Conjecture (Two Primes with given Sum)
  7. Circular primes less than n
  8. Check a number for Permutable Prime
  9. N expressed as sum of 4 prime numbers
  10. Check if a number can be written as a sum of ‘k’ prime numbers
  11. Express an odd number as sum of prime numbers
  12. Find the prime numbers which can written as sum of most consecutive primes
  13. Finding a Non Transitive Coprime Triplet in a Range

Quick Links:

• Recent Articles on Prime Number
• Practice problems on Mathematical algorithms