43,112,609

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.

43112609
List of numbers Integers ← 100 101 102 103 104 105 106 107 108 109
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	${\displaystyle {\stackrel {\delta \tau \iota \alpha }{\mathrm {M} }}}$͵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 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. ^[1] 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. ^[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 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) 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]

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

• George Woltman, Scott Kurowski, On the discovery of the 45th and 46th known Mersenne primes", Fibonacci Quarterly, vol. 46/47, no. 3, pp. 194–197, August 2008.