How to Identify a Prime Number Without a Computer
10 days ago
- #Prime Numbers
- #Mersenne Primes
- #Mathematics
- Édouard Lucas proved the 39-digit number 170,141,183,460,469,231,731,687,303,715,884,105,727 (2^127 - 1) is prime without computers.
- Prime numbers are divisible only by 1 and themselves and form the building blocks of mathematics.
- Mersenne primes have the form 2^p - 1, where p is a prime number, but not all such numbers are primes.
- Lucas developed the Lucas-Lehmer primality test to verify Mersenne primes, reducing the computational effort significantly.
- The test involves generating a sequence where the (p-2)nd term must be divisible by the Mersenne number for it to be prime.
- Finite number fields, studied by Galois, underpin the Lucas-Lehmer test, leveraging their symmetric properties when p is prime.
- The largest known prime as of October 2025 is 2^13,627,984 - 1, a Mersenne prime with over 41 million digits.