Q. Is 170141183460469231731687303715884105727 prime number

The number 170,141,183,460,469,231,731,687,303,715,884,105,727 is indeed a prime number, known as a Mersenne prime and specifically designated as M₁₂₇. This number equals 2¹²⁷ - 1, making it the largest Mersenne prime discovered before the advent of modern computing. Mersenne primes follow the form 2ᵖ - 1 where p itself must be prime, and they're particularly significant in number theory and have practical applications in computer science, including random number generation and cryptography.

When this prime was discovered in 1876 by Édouard Lucas, it held the record as the largest known prime number for several decades. The number contains 39 digits and demonstrates remarkable mathematical properties characteristic of Mersenne primes. Verifying such large numbers as prime requires specialized algorithms like the Lucas-Lehmer test, which is specifically designed for Mersenne numbers and is significantly more efficient than general primality tests. Today, while much larger Mersenne primes have been discovered through distributed computing projects, M₁₂₇ remains mathematically significant and is often used in computer systems—for instance, it represents the maximum value for a 127-bit signed integer in some computing contexts.