20210908, 04:27  #1970 
Jun 2015
Vallejo, CA/.
3×383 Posts 
Both forms of numbers exist,
The 3*2^{k}  1 are called Thabit primes and there are 64 of them known. https://oeis.org/A002235/b002235.txt Those of the Form 3*2^{k} + 1 are called Thabit of the second kind and 50 of them are known https://oeis.org/A002253/b002253.txt 
20210908, 05:06  #1971 
Mar 2006
Germany
2,999 Posts 
Should read "321 Prime Search i.e. 3*2^{k}±1" so the PrimeGrid Prime Search for both forms.

20210929, 09:48  #1972 
"Dmitrii"
Mar 2021
St.Petersburg,Russia
11 Posts 
3317*2^24159981 is prime! (727292 decimal digits)

20211015, 15:41  #1973  
Jun 2015
Vallejo, CA/.
47D_{16} Posts 
Quote:
.....................HERE IS THE OFFICIAL ANNOUNCEMENT PrimeGrid’s 321 Prime Search On 6 September 2021, 07:16:28 UTC, PrimeGrid’s 321 Prime Search found another Mega Prime: 3*2^{17748034}1 The prime is 5,342,692 digits long and will enter Chris Caldwell's “The Largest Known Primes Database” (http://primes.utm.edu/primes) ranked 18th overall. This is the largest known prime for the 3*2^n1 form. The discovery was made by Marc Wiseler of Ireland using an AMD Ryzen 9 5900X 12Core Processor with 16GB RAM, running Microsoft Windows 10 Professional x64 Edition. This computer took about 2 hours, 45 minutes to complete the primality test using LLR. Marc Wiseler is a member of Storm. The prime was verified on 6 September 2021, 11:47 UTC, by an Intel(R) Core(TM) i79800X CPU @ 3.80GHz with 32GB of RAM, running CentOS. This computer took 2 hours and 41 minutes to complete the primality test using LLR2. Credits for the discovery are as follows: 1. Marc Wiseler (Ireland), discoverer 2. PrimeGrid, et al. 3. Srsieve, sieving program developed by Geoff Reynolds 4. PSieve, sieving program developed by Ken Brazier and Geoff Reynolds 5. LLR, primality program developed by Jean Penné 6. LLR2, primality program developed by Pavel Atnashev Entry in “The Largest Know Primes Database” can be found here: https://primes.utm.edu/primes/page.php?id=132678 Using a single PC would have taken years to find this prime. So this timely discovery would not have been possible without the thousands of volunteers who contributed their spare CPU cycles. A special thanks to everyone who contributed their advice and/or computing power to the search  especially all the sievers who work behind the scenes to make a find like this possible. The 321 Prime Search will continue to seek even larger primes. To join the search please visit PrimeGrid: http://www.primegrid.com Last fiddled with by Dr Sardonicus on 20211015 at 16:02 Reason: formatting (superscript exponent) 

20211020, 13:23  #1974 
"Dmitrii"
Mar 2021
St.Petersburg,Russia
11 Posts 
3317*2^31171621 is prime! (938363 decimal digits)

20211027, 16:10  #1975 
"Dmitrii"
Mar 2021
St.Petersburg,Russia
11 Posts 
3317*2^25233661 is prime! (759613 decimal digits)

20211119, 12:46  #1976 
"W. Byerly"
Aug 2013
81*2^31743531
7×19 Posts 
81*2^23754471 is prime! (715083 digits)
81*2^27974431 is prime! (842117 digits) (New personal record ) 
20211120, 19:21  #1977 
Feb 2003
2^{7}×3×5 Posts 
233*2^28463921 (856852 digits)

20211204, 19:27  #1978 
Mar 2016
2^{2} Posts 
1005·2^3,349,0461 (1,008,167 digits)

20220326, 15:02  #1979 
Jun 2015
Vallejo, CA/.
10001111101_{2} Posts 
New 321 prime record
3 *2 ^{18924898} 1 5,696,990 Digits L5530 Mar 2022 (Unverified because InProcess) Congratulations to FMatillek Last fiddled with by rudy235 on 20220326 at 15:18 Reason: Typo 
20220331, 09:20  #1980 
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
3^{2}×1,117 Posts 
Just for kicks and giggles I tested a little bit of k=1965 (which happens to be my year) and found unlikely quickly
1965 · 2^{3322579}  1 (1,000,200 digits) and 1965 · 2^{3325639}  1 (1,001,121 digits). Just a curiosity. 
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