The number 2,147,483,647 (two billion one hundred forty-seven million four hundred eighty-three thousand six hundred forty-seven) is the eighth Mersenne prime, equal to 231 − 1. It is one of only four known double Mersenne primes.[1]
The primality of this number was proved by Leonhard Euler, who reported the proof in a letter to Daniel Bernoulli written in 1772.[2] Euler used trial division, improving on Cataldi's method, so that at most 372 divisions were needed.[3] The number 2,147,483,647 may have remained the largest known prime until 1876.[4]
Contents |
In 1811, Peter Barlow, not anticipating future interest in prime numbers, 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.[5]
He repeated this prediction in his 1814 work A New Mathematical and Philosophical Dictionary.[6][7]
The number 2,147,483,647 is also the maximum value for a 32-bit signed integer in computing. It is therefore the maximum value for variables declared as int in many programming languages running on popular CPUs, and the maximum possible score (or amount of money) for many video games. The appearance of the number often reflects an error, overflow condition, or missing value.[8] Similarly, "(214) 748-3647" is the sequence of digits represented as a United States phone number and is the most common phone number listed on web pages.[9]
The data type time_t, used on operating systems such as Unix, is a 32-bit signed integer counting the number of seconds since the start of the Unix epoch (midnight UTC of 1 January 1970).[10] The latest time that can be represented this way is 03:14:07 UTC on Tuesday, 19 January 2038 (corresponding to 2,147,483,647 seconds since the start of the epoch), so that systems using a 32-bit time_t type are susceptible to the Year 2038 problem.[11]