43,112,609 (forty-three million, one hundred twelve thousand, six hundred nine) is the

natural number In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...

following 43,112,608 and preceding 43,112,610.

In mathematics

43,112,609 is a prime number. Moreover, it is the exponent of the 47th

Mersenne prime 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 for some integer . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 1 ...

, equal to M_43,112,609 = 2^43,112,609 − 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 The Great Internet Mersenne Prime Search (GIMPS) is a collaborative project of volunteers who use freely available software to search for Mersenne prime numbers. GIMPS was founded in 1996 by George Woltman, who also wrote the Prime95 client and ...

. The 45th Mersenne prime, M_37,156,667 = 2^37,156,667 − 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, M_42,643,801 = 2^42,643,801 − 1, was discovered by Odd Magnar Strindmo, a GIMPS participant from

Norway Norway, officially the Kingdom of Norway, is a Nordic countries, Nordic country located on the Scandinavian Peninsula in Northern Europe. The remote Arctic island of Jan Mayen and the archipelago of Svalbard also form part of the Kingdom of ...

. 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. Having 12,837,064 decimal digits, it is only 141,125 digits, or 1.09%, shorter than M_43,112,609. 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) (also denoted \mathbb F_2, or \mathbb Z/2\mathbb Z) is the finite field with two elements. is the Field (mathematics), field with the smallest possible number of elements, and is unique if the additive identity and the multiplicative identity ...

found in 2016. and were the four largest in 2011. 43,112,609 is a

Sophie Germain prime In number theory, a prime number ''p'' is a if 2''p'' + 1 is also prime. The number 2''p'' + 1 associated with a Sophie Germain prime is called a . For example, 11 is a Sophie Germain prime and 2 × 11 +&nbs ...

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

In mathematics

References

Further reading