The number

Number

A number is a mathematical object used to count and measure. In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers....

 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

Mersenne prime

In mathematics, a Mersenne number, named after Marin Mersenne , is a positive integer that is one less than a power of two: M_p=2^p-1.\,...

, equal to 2³¹ − 1. It is one of only four known double Mersenne primes.

The primality

Primality test

A primality test is an algorithm for determining whether an input number is prime. Amongst other fields of mathematics, it is used for cryptography. Unlike integer factorization, primality tests do not generally give prime factors, only stating whether the input number is prime or not...

 of this number was proved by Leonhard Euler

Leonhard Euler

Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...

, who reported the proof in a letter to Daniel Bernoulli

Daniel Bernoulli

Daniel Bernoulli was a Dutch-Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family. He is particularly remembered for his applications of mathematics to mechanics, especially fluid mechanics, and for his pioneering work in probability and statistics...

 written in 1772. Euler used trial division

Trial division

Trial division is the most laborious but easiest to understand of the integer factorization algorithms. Its ease of implementation makes it a viable integer factorization option for devices with little available memory, such as graphing calculators....

, improving on Cataldi

Pietro Cataldi

Pietro Antonio Cataldi was an Italian mathematician. A citizen of Bologna, he taught mathematics and astronomy and also worked on military problems. His work included the development of continued fractions and a method for their representation. He was one of many mathematicians who attempted to...

's method, so that at most 372 divisions were needed. The number 2,147,483,647 may have remained the largest known prime

Largest known prime

The largest known prime number is the largest integer that is currently known to be a prime number.It was proven by Euclid that there are infinitely many prime numbers; thus, there is always a prime greater than the largest known prime. Many mathematicians and hobbyists search for large prime numbers...

 until 1876.

Barlow's prediction

In 1811, Peter Barlow, not anticipating future interest in prime number

Prime number

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...

s, 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.

He repeated this prediction in his 1814 work A New Mathematical and Philosophical Dictionary.

2147483647 in computing

The number 2,147,483,647 is also the maximum value for a 32-bit

32-bit

The range of integer values that can be stored in 32 bits is 0 through 4,294,967,295. Hence, a processor with 32-bit memory addresses can directly access 4 GB of byte-addressable memory....

 signed integer

Integer (computer science)

In computer science, an integer is a datum of integral data type, a data type which represents some finite subset of the mathematical integers. Integral data types may be of different sizes and may or may not be allowed to contain negative values....

 in computing

Computing

Computing is usually defined as the activity of using and improving computer hardware and software. It is the computer-specific part of information technology...

. It is therefore the maximum value for variables declared as int in many programming languages running on popular CPUs

Central processing unit

The central processing unit is the portion of a computer system that carries out the instructions of a computer program, to perform the basic arithmetical, logical, and input/output operations of the system. The CPU plays a role somewhat analogous to the brain in the computer. The term has been in...

, and the maximum possible score (or amount of money) for many video games

History of video game consoles (fifth generation)

The fifth-generation era refers to the computer and video games, video game consoles, and video game handhelds available at stores...

. The appearance of the number often reflects an error, overflow

Arithmetic overflow

The term arithmetic overflow or simply overflow has the following meanings.# In a computer, the condition that occurs when a calculation produces a result that is greater in magnitude than that which a given register or storage location can store or represent.# In a computer, the amount by which a...

 condition, or missing value. Similarly, "(214) 748-3647" is the sequence of digits represented as a United States phone number and is the most common phone number listed on web pages.

The data type time t, used on operating systems such as Unix

Unix

Unix is a multitasking, multi-user computer operating system originally developed in 1969 by a group of AT&T employees at Bell Labs, including Ken Thompson, Dennis Ritchie, Brian Kernighan, Douglas McIlroy, and Joe Ossanna...

, is a 32-bit signed integer counting the number of seconds since the start of the Unix epoch (midnight

Midnight

Midnight is the transition time period from one day to the next: the moment when the date changes. In the Roman time system, midnight was halfway between sunset and sunrise, varying according to the seasons....

 UTC of 1 January 1970). 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

Year 2038 problem

The year 2038 problem may cause some computer software to fail at some point near the year 2038...

.

External links

• Prime curios: 2147483647