2147483647

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2147483647
Cardinal two billion one hundred forty-seven million four hundred 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
Hexadecimal 7FFFFFFF16
Vigesimal 1DB1F92720
Base 36 ZIK0ZJ36

The number 2,147,483,647 (two billion one hundred forty-seven million four hundred eighty-three thousand six hundred forty-seven) is the eighth Mersenne prime, equal to 231 − 1. It is one of only four known double Mersenne primes.[1]

By 1772, Leonhard Euler had proved that 2,147,483,647 is 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 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]

Barlow's prediction[edit]

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, 2127 − 1, was found in 1876 by Lucas, and in 1883 Pervushin found the prime 261 − 1.

In computing[edit]

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 computers, 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]

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).[9] 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.[10]

See also[edit]

References[edit]

  1. ^ Weisstein, Eric W., "Double Mersenne Number", From MathWorld (A Wolfram Web Resource) .
  2. ^ Dunham, William (1999), Euler: The Master of Us All, Washington, DC: Mathematical Association of America, p. 4, ISBN 0-88385-328-0 .
  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 .
  4. ^ Caldwell, Chris (8 December 2009), The largest known prime by year .
  5. ^ Barlow, Peter (1811), An Elementary Investigation of the Theory of Numbers, London: J. Johnson & Co. 
  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 .
  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 .
  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. ^ "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. 
  10. ^ The Year-2038 Bug, archived from the original on 18 March 2009, retrieved 9 April 2009 .

External links[edit]