43,112,609

Last updated

43,112,609 (forty-three million, one hundred twelve thousand, six hundred nine) is the natural number following 43,112,608 and preceding 43,112,610.

Contents

43112609
Cardinal forty-three million one hundred twelve thousand six hundred nine
Ordinal 43112609th
(forty-three million one hundred twelve thousand six hundred ninth)
Factorization prime
Greek numeral ͵βχθ´
Roman numeral N/A
Binary 101001000111011000101000012
Ternary 100000101001010223
Senary 41400152256
Octal 2443542418
Duodecimal 1253151512
Hexadecimal 291D8A116

In mathematics

43,112,609 is a prime number. Moreover, it is the exponent of the 47th Mersenne prime, equal to M43,112,609 = 243,112,609  1, a prime number with 12,978,189 decimal digits. It was discovered on August 23, 2008 by Edson Smith, a volunteer of the Great Internet Mersenne Prime Search. [1] The 45th Mersenne prime, M37,156,667 = 237,156,667  1, was discovered two weeks later on September 6, 2008, marking the shortest chronological gap between discoveries of Mersenne primes since the formation of the online collaborative project in 1996. It was the first time since 1963 that two Mersenne primes were discovered less than 30 days apart from each other. Less than a year later, on June 4, 2009, the 46th Mersenne prime, M42,643,801 = 242,643,801  1, was discovered by Odd Magnar Strindmo, a GIMPS participant from Norway. [2] The result for this prime was first reported to the server in April 2009, but due to a bug, remained unnoticed for nearly two months. [3] Having 12,837,064 decimal digits, it is only 141,125 digits, or 1.09%, shorter than M43,112,609. These two Mersenne primes hold the record for the ones with the smallest ratio between their exponents.

43,112,609 is the degree of four of the seven largest primitive binary trinomials over GF(2) found in 2016. [4] and were the four largest in 2011. [5]

43,112,609 is a Sophie Germain prime, the largest of only eight known Mersenne prime indexes to have this property. [6]

43,112,609 is not a Gaussian prime, the largest of only 28 known Mersenne prime indexes to have this property. [7]

Related Research Articles

<span class="mw-page-title-main">Great Internet Mersenne Prime Search</span> Volunteer project using software to search for Mersenne prime numbers

The Great Internet Mersenne Prime Search (GIMPS) is a collaborative project of volunteers who use freely available software to search for Mersenne prime numbers.

In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form Mn = 2n − 1 for some integer n. They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17th century. If n is a composite number then so is 2n − 1. Therefore, an equivalent definition of the Mersenne primes is that they are the prime numbers of the form Mp = 2p − 1 for some prime p.

19 (nineteen) is the natural number following 18 and preceding 20. It is a prime number.

23 (twenty-three) is the natural number following 22 and preceding 24.

<span class="mw-page-title-main">Power of two</span> Two raised to an integer power

A power of two is a number of the form 2n where n is an integer, that is, the result of exponentiation with number two as the base and integer n as the exponent.

31 (thirty-one) is the natural number following 30 and preceding 32. It is a prime number.

In mathematics, a palindromic prime is a prime number that is also a palindromic number. Palindromicity depends on the base of the number system and its notational conventions, while primality is independent of such concerns. The first few decimal palindromic primes are:

127 is the natural number following 126 and preceding 128. It is also a prime number.

In number theory, a Wagstaff prime is a prime number of the form

Richard Peirce Brent is an Australian mathematician and computer scientist. He is an emeritus professor at the Australian National University. From March 2005 to March 2010 he was a Federation Fellow at the Australian National University. His research interests include number theory, random number generators, computer architecture, and analysis of algorithms.

In finite field theory, a branch of mathematics, a primitive polynomial is the minimal polynomial of a primitive element of the finite field GF(pm). This means that a polynomial F(X) of degree m with coefficients in GF(p) = Z/pZ is a primitive polynomial if it is monic and has a root α in GF(pm) such that is the entire field GF(pm). This implies that α is a primitive (pm − 1)-root of unity in GF(pm).

<span class="mw-page-title-main">1,000,000,000</span> Natural number

1,000,000,000 is the natural number following 999,999,999 and preceding 1,000,000,001. With a number, "billion" can be abbreviated as b, bil or bn.

100,000 (one hundred thousand) is the natural number following 99,999 and preceding 100,001. In scientific notation, it is written as 105.

1023 is the natural number following 1022 and preceding 1024.

<span class="mw-page-title-main">Largest known prime number</span>

The largest known prime number is 282,589,933 − 1, a number which has 24,862,048 digits when written in base 10. It was found via a computer volunteered by Patrick Laroche of the Great Internet Mersenne Prime Search (GIMPS) in 2018.

10,000,000 is the natural number following 9,999,999 and preceding 10,000,001.

100,000,000 is the natural number following 99,999,999 and preceding 100,000,001.

<span class="mw-page-title-main">Megaprime</span> Prime number with at least one million digits

A megaprime is a prime number with at least one million decimal digits.

References

  1. "GIMPS Discovers 45th and 46th Mersenne Primes, M43,112,609 is now the Largest Known Prime. Titanic Primes Raced to Win $100,000 Research Award". 2008-09-16. Retrieved 2020-06-04.
  2. "GIMPS Discovers 47th Mersenne Prime, M42,643,801 is newest, but not the largest, known Mersenne Prime". 2009-06-12. Retrieved 2009-06-04.
  3. "16987...14751 (12837064 digits)". Prime Curios!. February 5, 2013.
  4. Richard P. Brent, Paul Zimmermann, "Twelve new primitive binary trinomials", arXiv:1605.09213, 24 May 2016,
  5. Richard P. Brent, Paul Zimmermann, "The Great Trinomial Hunt", Notices of the American Mathematical Society, vol. 58, no. 2, pp. 233–239, February 2011.
  6. (sequence A065406 in the OEIS )
  7. (sequence A112634 in the OEIS )

Further reading