Seventeen or bust

Seventeen or Bust (“Seventeen or Failure”) is a project of voluntary calculations to find primes of the form k · 2 ⁿ + 1 for seventeen different values of k , which will prove that 78 557 is the minimum Sierpinski number . The project started in March 2002, in April 2016, after the loss of a data server, it was absorbed by the PrimeGrid project and became its subproject. By the end of 2016, of the seventeen k values that need to be checked, only five remained: 21 181, 22 699, 24 737, 55 459 and 67 607 ^[1] .

Content

1 History
2 Current Status
3 See also
4 notes
5 Links

History

In 1962, John Selfridge proved that 78,557 is the Sierpinski number. In addition, in 1967, he and Vaclav Sierpinski suggested that 78,557 is the lowest Sierpinski number. However, this assumption is still a hypothesis . To confirm it, it is necessary to prove that numbers less than 78 557 are not Sierpinski numbers, that is, for every odd number k < 78 557, you need to find the number n at which the value k · 2 ⁿ + 1 is a prime number . When the project started, it was already done for all values of k except seventeen, hence the name of the project - “Seventeen or failure” .

If the project manages to find primes of the form k · 2 ⁿ + 1 for each of the remaining values of k , then the Selfridge and Sierpinski hypothesis will be proved. However, it is possible that the hypothesis is false, and one (or even several) of the remaining numbers k is the Sierpinski number. In this case, the project participants will not be able to find a prime number of the form k · 2 ⁿ + 1, and sooner or later the project will be forced to stop. At the same time, the performed calculations cannot serve as evidence of the belonging of the problematic number k to the Sierpinski numbers — it will have to be proved by other methods. The project may also fail due to the fact that the minimum required value of n is so huge that it cannot be found with the modern development of computer technology within a reasonable time, although this option is unlikely and contradicts heuristic estimates of n .

Current Status

As of January 2019 ^[2] :

Found 12 of the required 17 primes.
The largest of the numbers found, 10223 · 2 ^31172165 + 1, takes 8th place among the largest known primes and at the same time is the largest known prime number that is not a Mersenne number ^[3] .

Seventeen k values, as well as the values of twelve primes found, are given in the table:

No.	k	n	Signs k2 ⁿ + 1	opening date	Who found
one	4847	3321063	999744	October 15, 2005	Richard Hassler
2	5359	5054502	1521561	December 6, 2003	Randy sundquist
3	10223	31172165	9383761	October 31, 2016 ^[4]	Péter szabolcs
four	19249	13018586	3918990	March 26, 2007	Konstantin Agafonov
5	21181	> 31625000	> 9520000	Search continues
6	22699	> 31625000	> 9520000	Search continues
7	24737	> 31625000	> 9520000	Search continues
8	27653	9167433	2759677	June 8, 2005	Derek gordon
9	28433	7830457	2357207	December 30, 2004	anonymous member
10	33661	7031232	2116617	October 30, 2007	Sturle sunde
eleven	44131	995972	299823	December 6, 2002	deviced (nickname)
12	46157	698207	210186	November 27, 2002	Stephen gibson
13	54767	1337287	402569	December 22, 2002	Peter Coels
fourteen	55459	> 31625000	> 9520000	Search continues
fifteen	65567	1013803	305190	December 3, 2002	James burt
16	67607	> 31625000	> 9520000	Search continues
17	69109	1157446	348431	December 7, 2002	Sean dimichele

Notes

↑ Seventeen or Bust: Project Stats Archived December 24, 2013 on the Wayback Machine
↑ Project Statistics Page Archived February 4, 2012 to Wayback Machine
↑ The one hundred largest largest primes
↑ Found one of the largest primes with more than 9 million characters

Links

official site of Seventeen or Bust

[1] Seventeen or Bust: Project Stats Archived December 24, 2013 on the Wayback Machine

[2] Project Statistics Page Archived February 4, 2012 to Wayback Machine

[3] The one hundred largest largest primes

[4] Found one of the largest primes with more than 9 million characters