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

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Eudy Silva

Joined: 26 Aug 17
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Message 154959 - Posted: 27 Mar 2022 | 17:07:21 UTC

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.

Margus

Joined: 5 Feb 18
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Message 154960 - Posted: 27 Mar 2022 | 18:29:51 UTC - in response to Message 154959.

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

Eudy Silva

Joined: 26 Aug 17
Posts: 1957
ID: 918937
Credit: 558,676,147
RAC: 159,627

Message 154961 - Posted: 27 Mar 2022 | 18:42:31 UTC - in response to Message 154960.

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

Joined: 13 Feb 12
Posts: 3119
ID: 130544
Credit: 2,188,565,631
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Message 154963 - Posted: 27 Mar 2022 | 20:12:23 UTC

Brilliant work. Someone put that on a t-shirt.

Eudy Silva

Joined: 26 Aug 17
Posts: 1957
ID: 918937
Credit: 558,676,147
RAC: 159,627

Message 154964 - Posted: 27 Mar 2022 | 20:28:39 UTC - in response to Message 154963.

Brilliant work. Someone put that on a t-shirt.

Thanks, Dave.

It'll have to be a very large t-shirt, lol

____________

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

JeppeSN

Joined: 5 Apr 14
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Message 154965 - Posted: 27 Mar 2022 | 20:35:06 UTC

I also like it.

You can also list all the k like this (every odd k not mentioned, up to 271,129, is proven non-Sierpiński by finding a specific n so that k*2^n+1 is prime):

+--------------------------------------------------+ | 21,181 conjectured non-Sierpiński composite | | 22,699 conjectured non-Sierpiński prime | | 24,737 conjectured non-Sierpiński composite | | 55,459 conjectured non-Sierpiński composite | | 67,607 conjectured non-Sierpiński prime | | | | 78,557 proven Sierpiński composite | | | | 79,309 conjectured non-Sierpiński prime | | 79,817 conjectured non-Sierpiński prime | | 91,549 conjectured non-Sierpiński composite | | 131,179 conjectured non-Sierpiński composite | | 152,267 conjectured non-Sierpiński prime | | 156,511 conjectured non-Sierpiński prime | | 163,187 conjectured non-Sierpiński composite | | 200,749 conjectured non-Sierpiński composite | | 209,611 conjectured non-Sierpiński composite | | 222,113 conjectured non-Sierpiński prime | | 225,931 conjectured non-Sierpiński prime | | 227,723 conjectured non-Sierpiński composite | | 229,673 conjectured non-Sierpiński composite | | 237,019 conjectured non-Sierpiński prime | | 238,411 conjectured non-Sierpiński composite | | | | 271,129 proven Sierpiński prime | +--------------------------------------------------+

If you start producing T-shirts, remember they can become obsolete if something is found in SoB, PSP, or ESP.

/JeppeSN

Eudy Silva

Joined: 26 Aug 17
Posts: 1957
ID: 918937
Credit: 558,676,147
RAC: 159,627

Message 154966 - Posted: 27 Mar 2022 | 20:42:53 UTC - in response to Message 154965.

That's nice, JeppeSN !

A T-shirt 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.

Eudy Silva

Joined: 26 Aug 17
Posts: 1957
ID: 918937
Credit: 558,676,147
RAC: 159,627

Message 154968 - Posted: 27 Mar 2022 | 21:03:39 UTC - in response to Message 154965.

Wjth the PG search project:

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

Rafael
Volunteer tester

Joined: 22 Oct 14
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Message 154969 - Posted: 27 Mar 2022 | 21:53:09 UTC - in response to Message 154965.

If you start producing T-shirts, 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 :)

Eudy Silva

Joined: 26 Aug 17
Posts: 1957
ID: 918937
Credit: 558,676,147
RAC: 159,627

Message 154970 - Posted: 27 Mar 2022 | 21:56:49 UTC - in response to Message 154969.

If you start producing T-shirts, 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

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

Message 154971 - Posted: 28 Mar 2022 | 5:44:04 UTC

Smart clothing...

Pavel Atnashev

Joined: 11 Aug 17
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Message 154972 - Posted: 28 Mar 2022 | 7:27:02 UTC

I also suggest this chart for those trying to understand SoB/PSP/ESP:
https://media.discordapp.net/attachments/770284055321116693/917126485515575376/FirstPrimesXSP-202705.png?width=725&height=452

It shows current expected number of primes for each k, compared to all other k's.

Dave

Joined: 13 Feb 12
Posts: 3119
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Credit: 2,188,565,631
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Message 154973 - Posted: 28 Mar 2022 | 7:50:22 UTC

That needs sticking on the homepage.

JeppeSN

Joined: 5 Apr 14
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Message 154974 - Posted: 28 Mar 2022 | 11:44:25 UTC - in response to Message 154968.

With fixes for 227,723 and 237,019:

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

/JeppeSN

Eudy Silva

Joined: 26 Aug 17
Posts: 1957
ID: 918937
Credit: 558,676,147
RAC: 159,627

Message 154975 - Posted: 28 Mar 2022 | 12:40:36 UTC - in response to Message 154974.

Thanks !
____________

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

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