The number 2,147,483,647 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 1867.[4] In 1814, Peter Barlow, not anticipating future interest in prime numbers, wrote (in A New Mathematical and Philosophical Dictionary):
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][6]
2147483647 in computing
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 for many video games. The appearance of the number often reflects an error, overflow condition, or missing value.[7]
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).[8] 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.[9]
References
- ^ Weisstein, Eric W. "Double Mersenne Number." From MathWorld--A Wolfram Web Resource.
- ^ William Dunham, Euler: The Master of Us All, Mathematical Association of America, 1999, page 4.
- ^ Walter Gautschi, Mathematics of computation, 1943-1993: a half-century of computational mathematics, American Mathematical Society, 1994, p 486.
- ^ Caldwell, Chris. The largest known prime by year.
- ^ Daniel Shanks, Solved and Unsolved Problems in Number Theory, American Mathematical Society, 2001, p 495.
- ^ Peter Barlow, A new mathematical and philosophical dictionary: comprising an explanation of terms and principles of pure and mixed mathematics, and such branches of natural philosophy as are susceptible of mathematical investigation, G. and S. Robinson, 1814: [1]
- ^ See, for example: [2]. A search for images on Google will find many with metadata values of 2147483647. This image, for example, claims to have been taken with a camera aperture of 2147483647.
- ^ "The Open Group Base Specifications Issue 6 IEEE Std 1003.1, 2004 Edition (definition of epoch)". IEEE and The Open Group. The Open Group. 2004. http://www.opengroup.org/onlinepubs/000095399/. Retrieved 7 March 2008.
- ^ The Year-2038 Bug, retrieved 9 April 2009.
External links
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