# 2,147,483,647

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The number **2,147,483,647** is the eighth Mersenne prime, equal to 2^{31} − 1. It is one of only four known double Mersenne primes.^{[1]}

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 forty-seventh) |

Factorization | 2147483647 |

Prime | Yes |

Greek numeral | ͵γχμζ´ |

Roman numeral | N/A |

Binary | 1111111111111111111111111111111_{2} |

Ternary | 12112122212110202101_{3} |

Octal | 17777777777_{8} |

Duodecimal | 4BB2308A7_{12} |

Hexadecimal | 7FFFFFFF_{16} |

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]}

## Barlow's predictionEdit

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 2

^{31}− 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., 2^{30}(2^{31}− 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 computingEdit

The number 2,147,483,647 (or hexadecimal 7FFFFFFF_{16}) 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, it was reported that PSY's music video "Gangnam Style" had exceeded the 32-bit integer limit for YouTube view count, necessitating YouTube to 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]}

## See alsoEdit

## ReferencesEdit

**^**Weisstein, Eric W. "Double Mersenne Number".*MathWorld*. Wolfram Research. Retrieved 29 January 2018.**^**Dunham, William (1999).*Euler: The Master of Us All*. Washington, DC: Mathematical Association of America. p. 4. ISBN 978-0-88385-328-3.**^**Gautschi, Walter (1994).*Mathematics of Computation, 1943–1993: A Half-Century of Computational Mathematics*. Proceedings of Symposia in Applied Mathematics.**48**. Providence, Rhode Island: American Mathematical Society. p. 486. ISBN 978-0-8218-0291-5.- ^
^{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. **^**Barlow, Peter (1811).*An Elementary Investigation of the Theory of Numbers*. London: J. Johnson & Co. p. 43.greatest.

**^**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.**^**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.**^**See, for example: [1]^{[permanent dead link]}. 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.**^**"Gangnam Style YouTube Overflow".**^**"'Gangnam Style' breaks YouTube". CNN.com. 3 December 2014. Retrieved 19 December 2014.**^**"No, Psy's 'Gangnam Style' Did Not Break YouTube Video Counter". Variety.com. 5 December 2014. Retrieved 8 August 2020.**^**"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.**^**"The Year-2038 Bug". Archived from the original on 18 March 2009. Retrieved 9 April 2009.

## External linksEdit