2,147,483,647

The number 2147483647 is the eighth Mersenne prime, equal to 2³¹ − 1. It is one of only four known double Mersenne primes.

The primality of this number was proven by Leonhard Euler, who reported the proof in a letter to Daniel Bernoulli written in 1772. Euler used trial division, improving on Pietro Cataldi's method, so that at most 372 divisions were needed. It thus improved upon the previous record-holding prime, 6,700,417 – also discovered by Euler – forty years earlier. The number 2,147,483,647 remained the largest known prime until 1867.

In computing, this number is the largest value that a signed 32-bit integer field can hold.

At the time of its discovery, 2,147,483,647 was the largest known prime number. In 1811, Peter Barlow wrote (in An Elementary Investigation of the Theory of Numbers):

Euler ascertained that 2³¹ − 1 = 2147483647 is a prime number; and this is the greatest at present known to be such, and, consequently, the last of the above perfect numbers [i.e., 2³⁰(2³¹ − 1)], which depends upon this, is the greatest perfect number known at present, and probably the greatest that ever will be discovered; for as they are merely curious, without being useful, it is not likely that any person will attempt to find one beyond it.

He repeated this prediction in his work from 1814, A New Mathematical and Philosophical Dictionary.

A larger prime, 67,280,421,310,721, was discovered in 1855 by Thomas Clausen, but he did not provide a proof. In 1867, the number 3,203,431,780,337 was proven to be prime.

The number 2,147,483,647 (or hexadecimal 7FFFFFFF₁₆) is the maximum positive value for a 32-bit signed binary integer in computing. It is therefore the maximum value for variables declared as integers (e.g., as int) in many programming languages.