Is 2147483647 a prime number or not?

Is 2,147,483,647 a Prime Number?

The number 2,147,483,647 holds a certain mystique, frequently popping up in computer science contexts and sparking curiosity about its mathematical properties. The most common question: is it a prime number?

Yes, 2,147,483,647 is indeed a prime number. More specifically, it’s a Mersenne prime, meaning it can be expressed in the form 2ⁿ – 1, where ‘n’ is also a prime number. In this case, n=31. This isn’t just any Mersenne prime; its historical significance is substantial. Leonhard Euler, one of history’s most prolific mathematicians, confirmed its primality in 1772, a significant feat given the computational limitations of the time. This discovery cemented its place as the eighth known Mersenne prime.

The number’s fame extends beyond its Mersenne primality. It’s also recognized as a double Mersenne prime. A double Mersenne prime takes the form 2^{(2p – 1)} – 1, where ‘p’ is also prime. For 2,147,483,647, p=5. This layered structure adds another dimension to its unique mathematical properties.

For years, 2,147,483,647 held the title of the largest known prime. Its prominence influenced early computing, frequently appearing as the maximum value for integers in various systems. This was a direct consequence of its size and the constraints of 32-bit computing, where it represented the largest signed integer that could be stored using two’s complement representation. While larger primes have since been discovered, its influence on early computing solidifies its continued relevance beyond purely theoretical number theory.

So, while it may not be the largest known prime anymore, 2,147,483,647 remains a fascinating number, a testament to Euler’s ingenuity, and a key figure in the intersection of mathematics and computing history. Its status as both a Mersenne prime and a double Mersenne prime distinguishes it within the vast landscape of prime numbers.