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 Batalov 2021-10-22 15:57

2 Attachment(s)
[QUOTE=pepi37;591322]If I may ask how many candidates remain after that ?[/QUOTE]
Sure.

 bur 2021-10-22 17:44

New Primorial Prime found

Recently a new -1 type primorial prime was found at PRPNet.

[URL="https://primes.utm.edu/primes/page.php?id=132758"]3267113# - 1[/URL]

It has 1418398 digits, making it the largest known one. The last -1 primorial prime was found more than 9 years ago, so this is quite the finding. The last +1 primorial prime hit is from 2001 btw. 20 years ago. :D

I think it's an interesting type of prime due to its involvement in Euclid's proof of the infinitude of primes. Not many people seem to hunt for them though and they seem somewhat scarce taking into account that N+-1 has lots of factors.

[color=red][B]MODERATOR NOTE:[/b] Moved to this thread, which already has [url=https://www.mersenneforum.org/showpost.php?p=588860&postcount=275]this post[/url] and several followups related to this number.[/color]

 paulunderwood 2021-10-22 18:03

[QUOTE=bur;591379]Recently a new -1 type primorial prime was found at PRPNet.

[URL="https://primes.utm.edu/primes/page.php?id=132758"]3267113# - 1[/URL]

It has 1418398 digits, making it the largest known one. The last -1 primorial prime was found more than 9 years ago, so this is quite the finding. The last +1 primorial prime hit is from 2001 btw. 20 years ago. :D

I think it's an interesting type of prime due to its involvement in Euclid's proof of the infinitude of primes. Not many people seem to hunt for them though and they seem somewhat scarce taking into account that N+-1 has lots of factors.[/QUOTE]

PRP tests for these are quite quick, compared to proofs. However they need generic modular reduction for Fermat PRP tests, whereas small-k Riesel and Proth prime run 4x (?) faster using a special mod.

The rarity of these numbers might put the next beyond the powers of Batalov-Propper.

[color=red][B]MODERATOR NOTE:[/b] Moved to this thread, which already has [url=https://www.mersenneforum.org/showpost.php?p=588860&postcount=275]this post[/url] and several followups related to this number.[/color]

 bur 2021-10-25 17:34

Thanks for moving the post.
[QUOTE]The rarity of these numbers might put the next beyond the powers of Batalov-Propper.[/QUOTE]Who knows, it's not like where GIMPS is currently at where 20M+ consecutive candidates are composite - at least I don't think so.

I always forget the estimate for the digit size of primorials but the FFT size remains very managable even up to 20,000,000# where it's 3M.

So if anyone was willing to put some larger ressources towards primorials or factorials, I'm pretty sure it'll yield some nice results before ending up in GIMPS waters.

 Dr Sardonicus 2021-10-25 20:53

[QUOTE=bur;591586]I always forget the estimate for the digit size of primorials but the FFT size remains very managable even up to 20,000,000# where it's 3M.[/QUOTE]The number of decimal digits in p[sub]k[/sub] is roughly p[sub]k[/sub]/ln(10). We have

? 3267113/log(10)
%1 = 1418889.1476543787883927627556683863802

As indicated above, 3267113# - 1 actually has 1418398 digits.

The estimate is a consequence of the Prime Number Theorem, which gives the asymptotic estimate

ln(p[sub]k[/sub]#) = ln(2) + ln(3) + ... + ln(p[sub]k[/sub]) ~ p[sub]k[/sub]

 paulunderwood 2022-01-10 19:42

288465! + 1

Congrats to RenĂ© Dohmen for the new [URL="https://primes.utm.edu/primes/page.php?id=133139"]factorial prime 288465! + 1[/URL] (1,449,771 decimal digits) :banana:

 unconnected 2022-01-10 22:07

Seems that this prime was found outside of [URL="http://prpnet.primegrid.com:12002/"]Factorial Prime Search[/URL] by Primegrid.

 paulunderwood 2022-01-10 22:39

[QUOTE=unconnected;597601]Seems that this prime was found outside of [URL="http://prpnet.primegrid.com:12002/"]Factorial Prime Search[/URL] by Primegrid.[/QUOTE]

Sniped! :snipe:

 rudy235 2022-01-10 23:30

Time to update
[url]https://oeis.org/A002982[/url]:smile:

 mathwiz 2022-01-11 01:32

[QUOTE=rudy235;597612]Time to update
[url]https://oeis.org/A002982[/url]:smile:[/QUOTE]

I think you mean [url]https://oeis.org/A002981[/url]?

 rudy235 2022-01-11 07:32

[QUOTE=mathwiz;597627]I think you mean [url]https://oeis.org/A002981[/url]?[/QUOTE]

Exactly! N! + 1