# 2,147,483,647

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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 forty-seventh) |

Factorization | 2147483647 |

Prime | Yes |

Greek numeral | ͵γχμζ´ |

Roman numeral | MMCXLVII CDLXXX MMMDCXLVII |

Binary | 1111111111111111111111111111111_{2} |

Ternary | 12112122212110202101_{3} |

Quaternary | 1333333333333333_{4} |

Quinary | 13344223434042_{5} |

Senary | 553032005531_{6} |

Octal | 17777777777_{8} |

Duodecimal | 4BB2308A7_{12} |

Hexadecimal | 7FFFFFFF_{16} |

Vigesimal | 1DB1F927_{20} |

Base 36 | ZIK0ZJ_{36} |

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]}

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 1867.^{[4]}

## Barlow's prediction[edit]

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 prime 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.^{[8]}

## In computing[edit]

The number 2,147,483,647 (or hexadecimal 7FFF,FFFF_{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;^{[9]} 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.^{[10]}^{[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 also[edit]

## References[edit]

**^**Weisstein, Eric W., "Double Mersenne Number",*From MathWorld*, A Wolfram Web Resource.**^**Dunham, William (1999),*Euler: The Master of Us All*, Washington, DC: Mathematical Association of America, p. 4, ISBN 0-88385-328-0.**^**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.**^**Caldwell, Chris (8 December 2009),*The largest known prime by year*.**^**Barlow, Peter (1811),*An Elementary Investigation of the Theory of Numbers*, London: J. Johnson & Co.**^**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, RI: American Mathematical Society, p. 495, ISBN 0-8218-2824-X.**^**Caldwell, Chris. "The Largest Known Prime by Year: A Brief History". Prime Pages. Retrieved January 20, 2016.**^**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.**^**"Gangnam Style YouTube Overflow".**^**"'Gangnam Style' breaks YouTube". CNN.com. 2014-12-03. Retrieved 2014-12-19.**^**"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.**^***The Year-2038 Bug*, archived from the original on 18 March 2009, retrieved 9 April 2009.