2147483647

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The number 2,147,483,647 (written out: two billion one hundred forty-seven million four hundred and eighty-three thousand six hundred and forty-seven ) is the eighth Mersenne prime ( prime of the form ) and thus equals 2 31  - 1. It is also the third of only four known double Mersenne primes .

2147483647
2147483647
presentation
dual 111 1111 1111 1111 1111 1111 1111 1111
Octal 177 7777 7777
Duodecimal 4 BB23 08A7
Hexadecimal 7FFF FFFF
Morse code · · - - - · - - - - · · · · - - · · · · · · - - - - · · · · - - - · · · · · · - - - · · · 
Mathematical properties
sign positive
parity odd
Factorization Prime number
Divider 1, 2147483647

discovery

In 1772 Leonhard Euler proved that 2,147,483,647 is a prime number . He wrote this in a letter to his physicist and mathematician colleague Daniel Bernoulli . He used the trial division , an improved variant of Pietro Cataldi's method , so that he had to divide a maximum of 372 times. This made it the highest prime number discovered up to that time and by far exceeded the previous record holder of 6,700,417, who had been discovered by Euler forty years earlier. The number remained the largest known prime until 1867 .

Barlow's prediction

In 1811, Peter Barlow , who refused to delve into prime numbers , wrote in An Elementary Investigation of the Theory of Numbers :

Euler ascertained that 2 31  - 1 = 2 147 483 647 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.
Euler found that 2 31  - 1 = 2 147 483 647 is a prime number; and that this is currently the largest of these, and accordingly the last of the above-mentioned perfect numbers [e.g. B. 2 30 (2 31  - 1)] based on this is the largest currently known perfect number, and probably the largest that will ever be discovered; and because they are only weird and not useful, it is unlikely that anyone will try to find a larger one.

In his work A New Mathematical and Philosophical Dictionary , published in 1814, Peter Barlow repeated his statement. Contrary to Barlow's prediction, however, larger prime numbers were discovered (some without evidence), for example 1851 (999 999 000 001) and 1855 (67 280 421 310 721). In 1867 it was also proven that 3,203,431,780,337 is a prime number.

32-bit integer limit for computers

2 147 483 647 (or hexadecimal 7FFF, FFFF 16 ) is the largest positive value that can be stored in a 32-bit signed integer . As a result, in many programming languages ​​it is the maximum value for variables that are defined as integers (e.g. as int) and the highest possible result in many fifth generation video games .

The use of the number often indicates an error, arithmetic overflow, or missing amount.

In December 2014, the number of call exceeded Psys video " Gangnam Style " on YouTube , the 32-bit -integer-limit of 2,147,483 647. This forced YouTube to the variable to a 64-bit to change -integer.

The time_t data type , used in operating systems such as Unix , is a signed integer that counts the number of seconds since Unix time began on January 1, 1970 at 00:00:00, and is often referred to as a 32-bit Integer implemented. The last time that can be displayed in this form is 03:14:07 UTC on January 19, 2038 because 2 147 483 647 seconds have passed since the count began. This means that systems using a 32-bit time_t type are prone to the year 2038 problem . Systems using a larger 64-bit type_t do not suffer from this limitation.

Web links

Individual evidence

  1. ^ Eric W. Weisstein: Double Mersenne Number from From MathWorld , A Wolfram Web Resource
  2. ^ William Dunham: The Master of Us All , p. 4, Mathematical Association of America , Washington, DC, 1999, ISBN 0-88385-328-0
  3. ^ Walter Gautschi: Mathematics of computation, 1943–1993: a half-century of computational mathematics , p. 486, from Proceedings of Symposia in Applied Mathematics , Volume 48, American Mathematical Society , Providence, RI, 1994, ISBN 0-8218- 0291-7
  4. ↑ Chris Caldwell: The largest known prime by year , December 8, 2009
  5. ↑ Peter Barlow: An Elementary Investigation of the Theory of Numbers , J. Johnson & Co., London, 1811
  6. ↑ Peter Barlow: 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 , G. and S. Robinson, London, 1814
  7. ^ Daniel Shanks: Solved and Unsolved Problems in Number Theory , p. 495, fourth edition, American Mathematical Society , Providence, RI, 2001, ISBN 0-8218-2824-X
  8. ↑ See for example here: http://publib.boulder.ibm.com/infocenter/iseries/v5r4/index.jsp?topic=/apis/fstat.htm . A Google search for images will find many with metadata values ​​of 2,147,483,647. This image, for example, claims to have been taken with a camera aperture of 2,147,483,647.
  9. ↑ Gangnam Style exceeds 32-bit limit . Archived from the original on December 3, 2014. Retrieved July 2, 2015.
  10. ↑ 'Gangnam Style' breaks YouTube (English) . CNN.com. December 3, 2014. Retrieved December 19, 2014.
  11. ↑ The Open Group Base Specifications Issue 6 IEEE Std 1003.1, 2004 Edition (definition of epoch) . In: IEEE and The Open Group . The Open Group . 2004. Retrieved March 7, 2008.
  12. ^ The Year-2038 Bug ( March 18, 2009 memento in the Internet Archive ), accessed April 9, 2009