Titanic prime number

The term titanic prime (English titanic prime (number) ) was purchased from Samuel Yates coined and refers to a prime number with at least 1,000 decimal places.
The smallest titanic prime numbers have exactly 1000 digits, are of the form and have the following : ${\displaystyle p=10^{999}+n}$ ${\displaystyle n}$

n = 7, 663, 2121, 2593, 3561, 4717, 5863, 9459, 11239, 14397, 17289, 18919, 19411, 21667, 25561, 26739, 27759, 28047, 28437, 28989, 35031, 41037, 41409, 41451, 43047, 43269, 43383, 50407, 51043, 52507, 55587, 59877, 61971, 62919, 63177 ... (sequence A074282 in OEIS )

The first two titanic prime numbers were discovered by Alexander Hurwitz on November 3, 1961 . It was the two Mersenne prime numbers with 1281 digits and 1332 digits. The primality of was calculated first on that day, but Hurwitz was the first to notice the output a few seconds before on the computer . This led to a brief discussion between Selfridge and Hurwitz about which prime number was discovered first. It is official . ${\displaystyle M_{4253}=2^{4253}-1}$ ${\displaystyle M_{4423}=2^{4423}-1}$ ${\displaystyle M_{4253}}$ ${\displaystyle M_{4423}}$ ${\displaystyle M_{4253}}$ ${\displaystyle M_{4423}}$

Someone who has discovered a titanic prime number is, according to Samuel Yates, a titan (English titan ).

species

Gigantic prime number

A gigantic prime (English gigantic prime (number) ) is a prime number of at least 10,000 decimal places. This name was first mentioned in the 1992 article Collecting gigantic and titanic primes by Samuel Yates.

The first gigantic prime was discovered on April 8, 1979 by Harry L. Nelson and David Slowinski . It was the Mersenne prime number with 13,395 digits. ${\displaystyle M_{44497}=2^{44497}-1}$

The smallest gigantic prime numbers have exactly 10,000 digits, are of the form and have the following : ${\displaystyle p=10^{9999}+n}$ ${\displaystyle n}$

n = 33603, 55377, 70999, 78571, 97779, 131673, 139579, 236761, 252391, 282097, 333811, 342037, 355651, 359931, 425427, 436363, 444129, 473143, 479859, 484423, 515787, 543447, 684423, 515787, 543447 709053, 709431, 780199, 781891, 788527, 813019… (Follow A142587 in OEIS )

Nowadays you can discover several (similarly small) gigantic prime numbers per day with a normal PC .

The number of newly found mega-prime numbers per year

Mega prime

A megaprime ( number) is a prime number with at least 1,000,000 decimal places.

The first megaprime was discovered on June 1, 1999 by Nayan Hajratwala. It was the Mersenne prime number with 2,098,960 digits. ${\displaystyle M_{6972593}=2^{6972593}-1}$

There are currently 741 megaprime numbers and 53 PRP numbers with at least one million digits known (as of July 9, 2020).

Prime prime number

A bevaprime ( number ) is a prime number with 1,000,000,000 decimal places. It is also called a gigantic prime number, but the risk of confusion with “gigantic prime number” is quite high in this case. The name was introduced by Chris Caldwell , but he has removed this name from his articles.

Although no prime numbers are known yet, it is known that almost all prime numbers are prime prime numbers. This is because there are an infinite number of prime numbers (see Euclid's theorem ), but only a finite number of these have fewer than a billion decimal places. So all “remaining” prime numbers must have more than a billion digits.

Prime number records

Below is a list of the smallest and largest (known) prime numbers of the above forms. However, some of them are numbers that have many properties of a prime number, but which one is not yet entirely sure whether they are actually prime numbers or "only" pseudo- prime numbers . Such "probable prime numbers" are called PRP numbers (as of August 1, 2020):

number	status	record	shape	Decimal places	Discovery date	Explorer	swell
${\displaystyle 10^{999}-6101}$	prim	largest non-titanic prime number	---	999	?	?
${\displaystyle 10^{999}+7}$	prim	smallest titanic prime number	titanic	1000	?	?
${\displaystyle 10^{9999}-11333}$	prim	largest titanic, but not gigantic prime number	titanic	9,999	?	?
${\displaystyle 10^{9999}+33603}$	prim	smallest gigantic prime number	gigantic	10,000	August 2003	Jens Franke , Thorsten Kleinjung, Tobias Wirth
${\displaystyle 10^{99999}-59511}$	PRP	largest PRP number with fewer than 100,000 digits	gigantic	99,999	July 2009	Patrick De Geest
${\displaystyle 10^{99999}+309403}$	PRP	smallest PRP number with at least 100,000 digits	gigantic	100,000	January 2004	Daniel Heuer
${\displaystyle 3139\cdot 2^{3321905}-1}$	prim	largest secured gigantic prime that is not a megaprime	gigantic	999.997	January 14, 2008	Richard Hassler
${\displaystyle 10^{999999}-172473}$	PRP	largest gigantic PRP number that is not megaprime	gigantic	999,999	December 2016	Patrick De Geest
${\displaystyle 10^{999999}+593499}$	PRP	smallest PRP number with at least 1,000,000 digits	Mega prime	1,000,000	February 2013	Peter Kaiser
${\displaystyle 3292665455999520712131951625894^{32768}+1}$	prim	smallest secured mega prime number	Megaprime, generalized Fermatsche prime number F 15 (3292665455999520712131951625894)	1,000,000	20th July 2020	Ivy F. Tennant
${\displaystyle 2^{32582657}-1}$	prim	largest known mega prime number with fewer than 10,000,000 digits	Megaprime, 44. Mersenne prime M 32582657	9,808,358	September 4, 2006	Curtis Cooper, Steven R. Boone
${\displaystyle 2^{37156667}-1}$	prim	smallest known mega prime number with at least 10,000,000 digits	Megaprime, 45th Mersenne prime M 37156667	11.185.272	September 6, 2008	Hans-Michael Elvenich
${\displaystyle 2^{82589933}-1}$	prim	largest known mega-prime	Mega prime number, possibly 51. Mersenne prime number M 82 589 933	24,862,048	December 21, 2018	Patrick Laroche

The next list shows the 10 largest prime numbers so far. Most of them are Mersenne prime numbers , all of them are mega prime numbers (as of August 1, 2020).

rank	Prime number	property	Decimal places	Discovery date	Explorer	swell
1.	${\displaystyle 2^{82589933}-1}$	possibly 51. Mersenne prime number ${\displaystyle M_{82589933}}$	24,862,048	December 21, 2018	Patrick Laroche
2.	${\displaystyle 2^{77232917}-1}$	possibly 50th Mersenne prime number ${\displaystyle M_{77232917}}$	23.249.425	January 3, 2018	Jonathan Pace
3.	${\displaystyle 2^{74207281}-1}$	possibly 49. Mersenne prime number ${\displaystyle M_{74207281}}$	22,338,618	19th January 2016	Curtis Cooper
4th	${\displaystyle 2^{57885161}-1}$	possibly 48. Mersenne prime number ${\displaystyle M_{57885161}}$	17,425,170	5th February 2013	Curtis Cooper
5.	${\displaystyle 2^{43112609}-1}$	47. Mersenne prime number ${\displaystyle M_{43112609}}$	12,978,189	August 23, 2008	Edson Smith
6th	${\displaystyle 2^{42643801}-1}$	46. ​​Mersenne prime number ${\displaystyle M_{42643801}}$	12,837,064	June 13, 2009	Odd Magnar Strindmo
7th	${\displaystyle 2^{37156667}-1}$	45. Mersenne prime number ${\displaystyle M_{37156667}}$	11.185.272	September 6, 2008	Hans-Michael Elvenich
8th.	${\displaystyle 2^{32582657}-1}$	44. Mersenne prime number ${\displaystyle M_{32582657}}$	09,808,358	September 4, 2006	Curtis Cooper, Steven R. Boone
9.	${\displaystyle 10223\cdot 2^{31172165}-1}$	largest Colbert number (proof that there is no Sierpiński number , see also Seventeen or Bust ) ${\displaystyle k=10223}$	09,383,761	October 31, 2016	Péter Szabolcs
10.	${\displaystyle 2^{30402457}-1}$	43. Mersenne prime number ${\displaystyle M_{30402457}}$	09,152,052	December 15, 2005	Curtis Cooper, Steven R. Boone

Web links

Chris K. Caldwell: Smallest Titanics of Special Forms .

swell

1. ↑ http://primes.utm.edu/glossary/page.php?sort=TitanicPrime
2. ↑ ^{a b c d} Chris K. Caldwell: The Largest Known Prime by Year: A Brief History. Retrieved August 1, 2020 . 
3. ↑ http://primes.utm.edu/bios/page.php?lastname=Woltman
4. ↑ http://primes.utm.edu/glossary/xpage/GiganticPrime.html
5. ↑ http://primes.utm.edu/glossary/xpage/Megaprime.html
6. ↑ 2 ^6972593 - 1 on Prime Pages
7. ↑ ^{a b c d e} List of the 5000 largest known prime numbers (English). Retrieved August 1, 2020 . 
8. ^ ^{A b c} Henri Lifchitz, Renaud Lifchitz: PRP Records - Probable Primes Top 10000. PRP Records, accessed August 1, 2020 . 
9. ↑ Chris K. Caldwell: The Largest Known Prime by Year: A Brief History. January 1, 2016, accessed July 29, 2019 . 
10. ↑ Boivin: 6101. Prime Pages, accessed August 1, 2020 . 
11. ↑ Neil Sloane : Numbers n search did 10 ^ 999 + n is a (Titanic) prime. OEIS , accessed August 1, 2020 . 
12. ↑ ^{a b} Patrick De Geest: A free forum for Gigantic Primes. World Of Numbers, accessed August 1, 2020 . 
13. ↑ Norman Luhn: 10000 ... 33603 (10000 digits). Prime Pages, accessed August 1, 2020 . 
14. ↑ Neil Sloane : Numbers n search did 10 ^ 9999 + n is a (gigantic) prime. OEIS , accessed August 1, 2020 . 
15. ^ ^{A b} Henri Lifchitz, Renaud Lifchitz: PRP Records - Probable Primes Top 10000 - page 18. PRP Records, accessed on August 1, 2020 . 
16. ↑ PRP Records 10 ^ 99999-59511 (English). Retrieved August 1, 2020 . 
17. ↑ doorman: 10000 ... 09403 (100000-digits). Prime Pages, accessed August 1, 2020 . 
18. ↑ PRP Records 10 ^ 99999 + 309403 (English). Retrieved August 1, 2020 . 
19. ↑ 3139 · 2 ^3321905-1 on Prime Pages
20. ↑ PRP Records 10 ^ 999999-172473 (English). Retrieved August 1, 2020 . 
21. ↑ Patrick De Geest: Search for the first PRP mega prime of the form 10 ^ 999999 + y. PRP Records, accessed August 1, 2020 . 
22. ↑ PRP Records 10 ^ 999999 + 593499 (English). Retrieved August 1, 2020 . 
23. ↑ 3292665455999520712131951625894 ³²⁷⁶⁸ + 1 on Prime Pages
24. ↑ 2 ^32582657 - 1 on Prime Pages
25. ↑ 2 ^37156667 - 1 on Prime Pages
26. ↑ 2 ^82589933 - 1 on Prime Pages
27. ↑ GIMPS: GIMPS Discovers Largest Known Prime Number: 2 ^82,589,933 -1. Mersenne Research, Inc., accessed August 1, 2020 .  
28. ↑ Chris K. Caldwell: The Top Twenty: Mersenne. Prime Pages, accessed August 1, 2020 . 
29. ↑ List of the 20 largest prime numbers. Retrieved August 1, 2020 . 
30. ↑ List of the 20 largest Mersenne prime numbers. Retrieved August 1, 2020 . 
31. ↑ 2 ^82589933 - 1 on Prime Pages
32. ↑ 2 ^77232917 - 1 on Prime Pages
33. ↑ 2 ^74207281-1 on Prime Pages
34. ↑ 2 ^57885161 - 1 on Prime Pages
35. ↑ 2 ^43112609-1 on Prime Pages
36. ↑ 2 ^42643801-1 on Prime Pages
37. ↑ 2 ^37156667 - 1 on Prime Pages
38. ↑ 2 ^32582657 - 1 on Prime Pages
39. ↑ 10223 · 2 ^31172165 - 1 on Prime Pages
40. ↑ 10223 · 2 ^31172165 - 1 on primegrid.com (PDF)
41. ↑ 2 ^30402457 - 1 on Prime Pages