Seventeen or Bust

Seventeen or Bust is a collaborative internet project that has set itself the task of solving the Sierpiński problem .

The SoB server has not been accessible since mid-April 2016 and the future of the basic project is therefore uncertain. His questions are probably also answered in the two Internet projects on the Prime Sierpiński problem and the extended Sierpiński problem .

Sierpiński problem

The problem is: “What is the smallest Sierpiński number?” John L. Selfridge showed in 1962 that 78557 is a Sierpiński number . However, it is not yet known whether 78557 is the smallest Sierpiński number. It is assumed, however, that it is the smallest Sierpiński number. However, 17 other numbers come into question, all of which would be smaller than 78557 and thus could claim the title of the smallest Sierpiński number. These are the following 17 numbers:

4847, 5359, 10223, 19249, 21181, 22699, 24737, 27653, 28433, 33661, 44131, 46157, 54767, 55459, 65567, 67607, 69109

goal of the project

The project started in March 2002. It aims to prove that 78557 is actually the smallest Sierpiński number. To do this, it has to show that there is at least one for all the other 17 numbers mentioned above , so that: is a prime number . If such a number is found, the corresponding number can not be a Sierpiński number, because a Sierpiński number must be a composite number for all of them . ${\displaystyle k}$ ${\displaystyle n>0}$ ${\displaystyle k\cdot 2^{n}+1}$ ${\displaystyle n}$ ${\displaystyle k}$ ${\displaystyle k\cdot 2^{n}+1}$ ${\displaystyle n\geq 1}$

For each of the above 17 values ​​for , the project looks for prime numbers of the form ${\displaystyle k}$

${\displaystyle k\cdot 2^{1}+1,k\cdot 2^{2}+1,\dotsc ,k\cdot 2^{n}+1,\dotsc }$

It uses Proth's theorem . When a suitable one has been found, one has found a Proth prime number and at the same time proven above all that it is not a Sierpiński number. If one of all 17 numbers above could be found, then it is proven that 78557 is actually the smallest Sierpiński number. ${\displaystyle n}$ ${\displaystyle k}$ ${\displaystyle n}$

Of course, it can also be that for one or even more of the above numbers there actually does not exist such that it is a prime number. In this case, the search for a prime number would of course take infinitely long with no prospect of success. There are reasons, however, that the claim “78557 is the smallest Sierpiński number” is correct. ${\displaystyle k}$ ${\displaystyle n}$ ${\displaystyle k\cdot 2^{n}+1}$

Current result of the search

“Seventeen or Bust” has now found at least one for 12 of the 17 remaining numbers that leads to a prime number. ${\displaystyle k}$ ${\displaystyle n}$

k	n	Places of k • 2 n +1	Date of discovery	Explorer
46.157	698.207	210.186	November 27, 2002	Stephen Gibson
65,567	1,013,803	305.190	December 3, 2002	James P. Burt
44.131	995.972	299,823	December 6, 2002	Anonymous
69,109	1,157,446	348,431	December 7, 2002	Sean DiMichele
54,767	1,337,287	402,569	December 22, 2002	Peter Coels
5,359	5,054,502	1,521,561	December 6, 2003	Randy Sundquist
28,433	7,830,457	2,357,207	December 30, 2004	Ars Technica Team Prime Rib
27,653	9,167,433	2,759,677	June 8, 2005	Derek Gordon
4,847	3,321,063	999,744	October 15, 2005	Richard Hassler
19,249	13,018,586	3,918,990	May 5, 2007	Konstantin Agafonov
33,661	7,031,232	2,116,617	October 17, 2007	Sturdy Sunde
10,223	31.172.165	9,383,761	October 31, 2016	Péter Szabolcs
21,181	> 29,500,000	> 8,880,389	(in progress)
22,699	> 29,500,000	> 8,880,389	(in progress)
24,737	> 29,500,000	> 8,880,389	(in progress)
55,459	> 29,500,000	> 8,880,389	(in progress)
67,607	> 29,500,000	> 8,880,389	(in progress)

Colbert numbers

The prime number found by the “Seventeen or Bust” project is the currently largest known prime that is not a Mersenne prime (as of November 14, 2016). The six prime numbers of the above list with over a million digits: ${\displaystyle 10223\cdot 2^{31172165}+1}$

${\displaystyle 10223\cdot 2^{31172165}+1,19249\cdot 2^{13018586}+1,27653\cdot 2^{9167433}+1,28433\cdot 2^{7830457}+1,33661\cdot 2^{7031232}+1\ }$ and ${\displaystyle 5359\cdot 2^{5054502}+1}$

is also called Colbert numbers (this is also the definition of Colbert numbers: prime numbers with over a million digits that are found when searching with “Seventeen or Bust”). They were named after the American comedian and satirist Stephen T. Colbert .

outlook into the future

For the final proof that 78557 is the smallest Sierpiński number, it has to be shown that for the following there is at least one such that it is a prime number: ${\displaystyle k}$ ${\displaystyle n}$ ${\displaystyle k\cdot 2^{n}+1}$

${\displaystyle k=21181,22699,24737,55459,67607}$

It is assumed that at least one will actually be found for each of the above five at some point . The prime numbers found in this way will have over a million digits and are therefore also called Colbert numbers. One can also assume that the largest of the prime numbers found in this way is larger than all currently known prime numbers. ${\displaystyle k}$ ${\displaystyle n}$

Prime-Sierpiński problem

The possibly smallest Sierpiński number is a composite number. ${\displaystyle k=78557=17\cdot 4621}$

In 1976 Nathan Mendelsohn (1917–2006) proved that the prime number is also a Sierpiński number. This is currently the second smallest known Sierpiński number and the smallest known prime Sierpiński number . ${\displaystyle k=271129}$

The Prime-Sierpiński problem deals with whether the smallest prime Sierpiński number is actually . To check this, the following 9 prime numbers have to be checked (whereby the first two numbers already appear in the above problem) (as of December 31, 2019): ${\displaystyle k=271129}$

k = 22699, 67607, 79309, 79817, 152267, 156511, 222113, 225931, 237019

The Internet project “ Prime Sierpinski Project ” has been dealing with this question since January 1st, 2004.

Extended Sierpiński problem

The extended Sierpiński problem deals with whether the second smallest Sierpiński number is actually . In order to check this, in addition to the 9 prime numbers mentioned above, the following 12 composite numbers must also be checked (whereby the first three numbers already appear in the original problem) (as of December 31, 2019): ${\displaystyle k=271129}$

k = 21181, 24737, 55459, 91549, 131179, 163187, 200749, 202705, 209611, 227723, 229673, 238411

See also

• PrimeGrid - Internet search for record prime numbers

Individual evidence

1. ^ Louie Helm & David Norris: Seventeen or Bust. Accessed December 7, 2015 (English, project homepage). 
2. ↑ Michael Goetz: Re: Server down? .
3. ↑ Sierpinski problem. Mersennewiki, accessed December 7, 2015 (proof that k = 78557 is a Sierpinski number). 
4. ↑ Chris Caldwell: Sierpinski number. The Prime Glossary, accessed on December 7, 2015 (reasons why k = 78,557 is the smallest Sierpinski number). 
5. ^ Louie Helm & David Norris: Seventeen or Bust. (No longer available online.) Archived from the original on February 2, 2013 ; accessed on December 7, 2015 (English, current status of the project). Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice.   
6. ^ Weisstein, Eric W .: Sierpiński Number of the Second Kind. Wolfram MathWorld, accessed on December 7, 2015 (English, current status of the project). 
7. ↑ Chris K. Caldwell: Seventeen or Bust. Prime Pages, accessed December 7, 2015 (current status of the project). 
8. ↑ Chris K. Caldwell: Largest Known Primes. Prime Pages, accessed on November 14, 2016 (the 20 largest known prime numbers). 
9. ^ Helm, Louis: Colbert Number. Wolfram MathWorld, accessed November 14, 2016 . 
10. ↑ ^{a b} Chris K. Caldwell: Colbert number. Prime Pages, accessed December 7, 2015 . 
11. ↑ James Grime and Brady Haran : 78557 and Proth Primes - Numberphile. In: YouTube video. Numberphile, November 13, 2017, accessed November 14, 2017 . 
12. ^ Nathan S. Mendelsohn: The equation φ (x) = k . Math. Mag. 49, 1976, p. 37-39 .
13. ↑ ^{a b} Wilfrid Keller: The Sierpiński Problem: Definition and Status. Prothsearch, accessed on December 31, 2019 (English, extended Sierpiński problem).  
14. ^ Prime Sierpinski Project. rechenkraft.net, accessed on December 31, 2019 . 
15. ↑ Rytis Slatkevičius: Welcome to the Extended Sierpinski problem. PrimeGrid, accessed December 7, 2015 (English, extended Sierpiński problem). 

Web links

• Chris K. Caldwell: Riesel Sieve Project. Prime Pages, accessed December 7, 2015 (a related internet project for numbers of the form k · 2 ⁿ −1). 
• Louie Helm & David Norris: Seventeen or Bust. Accessed December 7, 2015 (English, project homepage).