# 2,147,483,647

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]

2147483647
Cardinaltwo billion one hundred forty-seven million four hundred eighty-three thousand six hundred forty-seven
Ordinal2147483647th
(two billion one hundred forty-seven million four hundred eighty-three thousand six hundred forty-seventh)
Factorizationprime
Prime105,097,565th
Greek numeral${\displaystyle {\stackrel {\kappa \alpha \delta \psi \mu \eta }{\mathrm {M} }}}$͵γχμζ´
Roman numeralN/A
Binary11111111111111111111111111111112
Ternary121121222121102021013
Octal177777777778
Duodecimal4BB2308A712
Hexadecimal7FFFFFFF16
By 1772, Leonhard Euler had proven that 2,147,483,647 is a prime.

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 Pietro 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 1867.[4]

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

## Barlow's prediction

At the time of its discovery, 2,147,483,647 was the largest known prime number. In 1811, Peter Barlow, not anticipating future interest in perfect 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.[4]

## In computing

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. The appearance of the number often reflects an error, overflow condition, or missing value.[8] In December 2014, it was reported that PSY's music video "Gangnam Style" had exceeded the 32-bit integer limit for YouTube's view counter, necessitating that YouTube upgrade the counter to a 64-bit integer.[9][10] In fact, this was an "easter egg" joke shared by the company, as Google had reportedly switched to a 64-bit integer several months prior.[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]

On 1 January 2022, a bug was reported for Microsoft Exchange systems where email delivery would fail. An internal malware scanner (enabled by default since 2013) used the date and time as a signed 32-bit integer. The integer would change during the new year to 2,201,010,001 (with the first two digits representing the year), surpassing the maximum value for this data type.[14]

## In video games

The number 2,147,483,647 is often used as a hard limit for various statistics in video games, such as points or money. Going over this limit by legitimate means, or by modding or hacking the game, results in many different outcomes. The most common outcome is the number going into the negatives; thus the limit being extended to 4,294,967,295, also known as the unsigned 32-bit integer limit. Another potential outcome is the game crashing, thus meaning there was no failsafe implemented in the event someone reached the 32-bit limit. A notorious example of this is in the 2001 racing video game Gran Turismo 3: A-Spec.

## References

1. ^ Weisstein, Eric W. "Double Mersenne Number". MathWorld. Wolfram Research. Retrieved 29 January 2018.
2. ^ Dunham, William (1999). Euler: The Master of Us All. Washington, DC: Mathematical Association of America. p. 4. ISBN 978-0-88385-328-3.
3. ^ Gautschi, Walter (1994). Mathematics of Computation, 1943–1993: A Half-Century of Computational Mathematics. Proceedings of Symposia in Applied Mathematics. Vol. 48. Providence, Rhode Island: American Mathematical Society. p. 486. ISBN 978-0-8218-0291-5.
4. ^ a b Caldwell, Chris (8 December 2009). "The Largest Known Prime by Year: A Brief History". The Prime Pages. University of Tennessee at Martin. Retrieved 29 January 2018.
5. ^ Barlow, Peter (1811). An Elementary Investigation of the Theory of Numbers. London: J. Johnson & Co. p. 43. greatest.
6. ^
7. ^ Shanks, Daniel (2001). Solved and Unsolved Problems in Number Theory (4th ed.). Providence, Rhode Island: American Mathematical Society. p. 495. ISBN 978-0-8218-2824-3.
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". Archived from the original on 23 December 2017.
10. ^ "'Gangnam Style' breaks YouTube". CNN. 3 December 2014. Retrieved 19 December 2014.
11. ^ "No, Psy's 'Gangnam Style' Did Not Break YouTube Video Counter". Variety.com. 5 December 2014. Retrieved 8 August 2020.
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. Archived from the original on 19 December 2008. Retrieved 7 March 2008.
13. ^ "The Year-2038 Bug". Archived from the original on 18 March 2009. Retrieved 9 April 2009.
14. ^ Abrams, Lawrence. "Microsoft Exchange year 2022 bug in FIP-FS breaks email delivery". Bleeping Computer. Retrieved 2 January 2022.