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drummerslowrise

Message boards :
Number crunching :
SOB, PSP and ESP at a glance
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Some time ago I got to the following while trying to better understand SOB, PSP, ESP (and TRP also).
Hope it is useful and correct :)
This message represents the situation as of 2022/03/27.
Numbers of the form k*2^n + 1.
A Sierpinski number is a value of k such that k*2^n + 1 is composite for any/every value of n you choose.
It has been proven that 78,557*2^n + 1 is composite for every n ≥ 1.
So, 78,557 is a Sierpinski number.
Is 78,557 the smallest Sierpinski number ?
To prove this is the case, we have to test all odd values of k below 78,557 and
show that for each k, at least one value of n makes k*2^n + 1 prime. As such, this particular k can not be a Sierpinski number.
This is the Seventeen or Bust project (SOB project).
It has been proven that 271,129*2^n + 1 is composite for every n ≥ 1.
So, 271,129 is a Sierpinski number. But, as 271,129 is also a prime number, it is a prime Sierpinski number.
Is 271,129 the smallest prime Sierpinski number ?
To prove this is the case, we have to test all odd prime values of k below 78,557 and
show that for each prime value of k at least one value of n makes k*2^n + 1. As such, this particular prime k can not be a Sierpinski number.
This is the Prime Sierpinski Problem (PSP project).
In case 78,557 is indeed the smallest Sierpinski number and 271,129 is indeed the smallest prime Sierpinski number, what about the k values between them ?
Is 271,129 the second smallest Sierpinski number ?
Or, putting it in another way, is there any other value of k between 78,557 (the smallest Sierpinski number) and 271,129 (the smallest prime Sierpinski number) that is also a Sierpinski number ?
To prove there is none, we have to test all odd values of k between 78,557 and 271,129 and find an 'n' that makes k*2^n + 1 prime.
Note that just composite odd k values need to be tested, since there must be no prime k Sierpinski numbers below 271,129, as 271,129 is the smallest prime Sierpinski number.
This is the Extended Sierpinski Problem (ESP project).
(PSP, 8 k left)
(SOB, 5 k left) Smallest
Smallest Prime
Sierpinski Sierpinski
number ? number ?
 
Number line 78,557 271,129
+++++++++++++++
21,181 22,699 24,737 55,459 67,607 79,309 79,817 152,267 156,511 168,451 222,113 225,931 237,019
____________________________ ________________________________________________________________________________________
odd k values to be tested to prove odd prime k values to be tested to prove 271,129 is the smallest prime Sierpinski number
78,557 is the smallest Sierpinski number (PSP)
(SOB)
(ESP, 8 k left)
91,549 131,179 163,187 200,749 209,611 227,723 229,673 238,411
________________________________________________________________________________
odd composite k values to be tested to prove there are no other Sierpinski numbers between 78,557 and 271,129
(so, 271,129, the smallest prime Sierpinski number, is also the second smallest Sierpinski number)
(ESP)
If you change "k*2^n + 1" to 'minus one', you get Riesel numbers: values of k that make k*2^n  1 composite for all n.
It has been proven that 509,203*2^n  1 is composite for every n ≥ 1.
So, 509,203 is a Riesel number.
Is 509,203 the smallest Riesel number ?
To prove that is the case, we have to test all odd values of k below 509,203 and
show that for each k, at least one value of n makes k*2^n  1 prime. As such, this particular k can not be a Riesel number.
This is the Riesel problem (TRP project)
____________
"Accidit in puncto, quod non contingit in anno."
Something that does not occur in a year may, perchance, happen in a moment.  


I think, k=168 451 in PSP is eliminated: http://www.primegrid.com/stats_psp_llr.php  


I think, k=168 451 in PSP is eliminated: http://www.primegrid.com/stats_psp_llr.php
Thanks for the correction !
Then we have
(PSP, 7 k left)
(SOB, 5 k left) Smallest
Smallest Prime
Sierpinski Sierpinski
number ? number ?
 
Number line 78,557 271,129
++++++++++++++
21,181 22,699 24,737 55,459 67,607 79,309 79,817 152,267 156,511 222,113 225,931 237,019
____________________________ __________________________________________________________________________________
odd k values to be tested to prove odd prime k values to be tested to prove 271,129 is the smallest prime Sierpinski number
78,557 is the smallest Sierpinski number (PSP)
(SOB)
(ESP, 8 k left)
91,549 131,179 163,187 200,749 209,611 227,723 229,673 238,411
________________________________________________________________________________
odd composite k values to be tested to prove there are no other Sierpinski numbers between 78,557 and 271,129
(so, 271,129, the smallest prime Sierpinski number, is also the second smallest Sierpinski number)
(ESP)
____________
"Accidit in puncto, quod non contingit in anno."
Something that does not occur in a year may, perchance, happen in a moment.  

Dave Send message
Joined: 13 Feb 12 Posts: 3119 ID: 130544 Credit: 2,188,565,631 RAC: 242,786

Brilliant work. Someone put that on a tshirt.  


Brilliant work. Someone put that on a tshirt.
Thanks, Dave.
It'll have to be a very large tshirt, lol
____________
"Accidit in puncto, quod non contingit in anno."
Something that does not occur in a year may, perchance, happen in a moment.  


I also like it.
You can also list all the k like this (every odd k not mentioned, up to 271,129, is proven nonSierpiński by finding a specific n so that k*2^n+1 is prime):
++
 21,181 conjectured nonSierpiński composite 
 22,699 conjectured nonSierpiński prime 
 24,737 conjectured nonSierpiński composite 
 55,459 conjectured nonSierpiński composite 
 67,607 conjectured nonSierpiński prime 
 
 78,557 proven Sierpiński composite 
 
 79,309 conjectured nonSierpiński prime 
 79,817 conjectured nonSierpiński prime 
 91,549 conjectured nonSierpiński composite 
 131,179 conjectured nonSierpiński composite 
 152,267 conjectured nonSierpiński prime 
 156,511 conjectured nonSierpiński prime 
 163,187 conjectured nonSierpiński composite 
 200,749 conjectured nonSierpiński composite 
 209,611 conjectured nonSierpiński composite 
 222,113 conjectured nonSierpiński prime 
 225,931 conjectured nonSierpiński prime 
 227,723 conjectured nonSierpiński composite 
 229,673 conjectured nonSierpiński composite 
 237,019 conjectured nonSierpiński prime 
 238,411 conjectured nonSierpiński composite 
 
 271,129 proven Sierpiński prime 
++
If you start producing Tshirts, remember they can become obsolete if something is found in SoB, PSP, or ESP.
/JeppeSN  


That's nice, JeppeSN !
A Tshirt with the PG logo would be cool.
____________
"Accidit in puncto, quod non contingit in anno."
Something that does not occur in a year may, perchance, happen in a moment.  


Wjth the PG search project:
+++++
 k  status  k primality  search project 
+++++
 21,181  conjectured nonSierpiński  composite  SOB 
 22,699  conjectured nonSierpiński  prime  SOB 
 24,737  conjectured nonSierpiński  composite  SOB 
 55,459  conjectured nonSierpiński  composite  SOB 
 67,607  conjectured nonSierpiński  prime  SOB 
+++++
 78,557  proven Sierpiński  composite  
+++++
 79,309  conjectured nonSierpiński  prime  PSP 
 79,817  conjectured nonSierpiński  prime  PSP 
 91,549  conjectured nonSierpiński  composite  ESP 
 131,179  conjectured nonSierpiński  composite  ESP 
 152,267  conjectured nonSierpiński  prime  PSP 
 156,511  conjectured nonSierpiński  prime  PSP 
 163,187  conjectured nonSierpiński  composite  ESP 
 200,749  conjectured nonSierpiński  composite  ESP 
 209,611  conjectured nonSierpiński  composite  ESP 
 222,113  conjectured nonSierpiński  prime  PSP 
 225,931  conjectured nonSierpiński  prime  PSP 
 227,723  conjectured nonSierpiński  composite  PSP 
 229,673  conjectured nonSierpiński  composite  ESP 
 237,019  conjectured nonSierpiński  prime  ESP 
 238,411  conjectured nonSierpiński  composite  ESP 
+++++
 271,129  proven Sierpiński  prime  
+++++
____________
"Accidit in puncto, quod non contingit in anno."
Something that does not occur in a year may, perchance, happen in a moment.  

RafaelVolunteer tester
Send message
Joined: 22 Oct 14 Posts: 909 ID: 370496 Credit: 503,312,821 RAC: 129,606

If you start producing Tshirts, remember they can become obsolete if something is found in SoB, PSP, or ESP.
/JeppeSN
Make them out of verlcro straps so you can remove candiadates as they get proven :)  


If you start producing Tshirts, remember they can become obsolete if something is found in SoB, PSP, or ESP.
/JeppeSN
Make them out of verlcro straps so you can remove candiadates as they get proven :)
😂
____________
"Accidit in puncto, quod non contingit in anno."
Something that does not occur in a year may, perchance, happen in a moment.  

Dave Send message
Joined: 13 Feb 12 Posts: 3119 ID: 130544 Credit: 2,188,565,631 RAC: 242,786

Smart clothing...  


I also suggest this chart for those trying to understand SoB/PSP/ESP:
https://media.discordapp.net/attachments/770284055321116693/917126485515575376/FirstPrimesXSP202705.png?width=725&height=452
It shows current expected number of primes for each k, compared to all other k's.  

Dave Send message
Joined: 13 Feb 12 Posts: 3119 ID: 130544 Credit: 2,188,565,631 RAC: 242,786

That needs sticking on the homepage.  


With fixes for 227,723 and 237,019:
+++++
 k  status  k primality  search project 
+++++
 21,181  conjectured nonSierpiński  composite  SOB 
 22,699  conjectured nonSierpiński  prime  SOB 
 24,737  conjectured nonSierpiński  composite  SOB 
 55,459  conjectured nonSierpiński  composite  SOB 
 67,607  conjectured nonSierpiński  prime  SOB 
+++++
 78,557  proven Sierpiński  composite  
+++++
 79,309  conjectured nonSierpiński  prime  PSP 
 79,817  conjectured nonSierpiński  prime  PSP 
 91,549  conjectured nonSierpiński  composite  ESP 
 131,179  conjectured nonSierpiński  composite  ESP 
 152,267  conjectured nonSierpiński  prime  PSP 
 156,511  conjectured nonSierpiński  prime  PSP 
 163,187  conjectured nonSierpiński  composite  ESP 
 200,749  conjectured nonSierpiński  composite  ESP 
 209,611  conjectured nonSierpiński  composite  ESP 
 222,113  conjectured nonSierpiński  prime  PSP 
 225,931  conjectured nonSierpiński  prime  PSP 
 227,723  conjectured nonSierpiński  composite  ESP 
 229,673  conjectured nonSierpiński  composite  ESP 
 237,019  conjectured nonSierpiński  prime  PSP 
 238,411  conjectured nonSierpiński  composite  ESP 
+++++
 271,129  proven Sierpiński  prime  
+++++
/JeppeSN  


Thanks !
____________
"Accidit in puncto, quod non contingit in anno."
Something that does not occur in a year may, perchance, happen in a moment.  

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Message boards :
Number crunching :
SOB, PSP and ESP at a glance 