# 2147483647 (number)

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2147483647
Cardinal two billion, one hundred and forty-seven million, four hundred and eighty-three thousand, six hundred and forty-seven
Ordinal 2147483647th
(two billion one hundred forty-seven million four hundred eighty-three thousand six hundred and forty-seventh)
Factorization 2147483647
Prime Yes
Roman numeral N/A
Binary 11111111111111111111111111111112
Ternary 121121222121102021013
Quaternary 13333333333333334
Quinary 133442234340425
Senary 5530320055316
Octal 177777777778
Duodecimal 4BB2308A712
Vigesimal 1DB1F92720
Base 36 ZIK0ZJ36

The number 2,147,483,647 (two billion, one hundred and forty-seven million, four hundred and eighty-three thousand, six hundred and 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 proven 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] 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 1855.[4]

## Barlow's prediction

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]

In fact a larger prime was discovered in 1855 by Thomas Clausen (67,280,421,310,721), though a proof was not provided. Furthermore, 3,203,431,780,337 was proven to be prime in 1867.

## In computing

The number 2,147,483,647 (or hexadecimal 7FFF,FFFF16) 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, and the maximum possible score, money, etc. for many video games. The appearance of the number often reflects an error, overflow condition, or missing value.[8] In December 2014, Google initially claimed that PSY's music video "Gangnam Style" exceeded the 32-bit integer limit for YouTube view count, necessitating YouTube to upgrade the variable to a 64-bit integer.[9][10] Google later admitted that this was a joke.[11]

The data type time_t, used on operating systems such as Unix, is a signed integer counting the number of seconds since the start of the Unix epoch (midnight UTC of 1 January 1970), and is often implemented as a 32-bit integer.[12] The latest time that can be represented in this form is 03:14:07 UTC on Tuesday, 19 January 2038 (corresponding to 2,147,483,647 seconds since the start of the epoch). This means that systems using a 32-bit `time_t` type are susceptible to the Year 2038 problem.[13] (Systems employing a wider 64-bit time_t type do not suffer from this limitation.)

## References

1. Weisstein, Eric W., "Double Mersenne Number", From MathWorld, A Wolfram Web Resource<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>.
2. Dunham, William (1999), Euler: The Master of Us All, Washington, DC: Mathematical Association of America, p. 4, ISBN 0-88385-328-0<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>.
3. Gautschi, Walter (1994), Mathematics of computation, 1943-1993: a half-century of computational mathematics, Proceedings of Symposia in Applied Mathematics, 48, Providence, RI: American Mathematical Society, p. 486, ISBN 0-8218-0291-7<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>.
4. Caldwell, Chris (8 December 2009), The largest known prime by year<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>.
5. Barlow, Peter (1811), An Elementary Investigation of the Theory of Numbers, London: J. Johnson & Co.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
6. Barlow, Peter (1814), 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, London: G. and S. Robinson<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>.
7. Shanks, Daniel (2001), Solved and Unsolved Problems in Number Theory (4th ed.), Providence, RI: American Mathematical Society, p. 495, ISBN 0-8218-2824-X<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>.
8. See, for example: [1]. 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.
9. "Gangnam Style YouTube Overflow".<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
10. "'Gangnam Style' breaks YouTube". http://www.cnn.com/. CNN.com. 3 December 2014. Retrieved 19 December 2014. External link in `|website=` (help)<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
11. "Gangnam Style' busts YouTube's view counter? Not so fast'".<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
12. "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. Retrieved 7 March 2008.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
13. The Year-2038 Bug, archived from the original on 18 March 2009, retrieved 9 April 2009 Unknown parameter `|deadurl=` ignored (help)<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>.