577294575*2^n+1

[roger]

Has this k been reserved by someone for the plus side? I found a couple proth primes on google relating to it, but no reservation status.

I've been working on it for a while, and am coming up to n=250k now.

I really hope this work hasn't already been done.

[VBCurtis]

The only place I know of coordinating proth prime searching is www.prothsearch.net. Mark only coordinates up to k=1200 as far as I can tell. If you find another location with organization info, please post it here.
-Curtis

[Kosmaj]

g42 has been testing this K since 1999 and found many primes:

rank           description              digits  who year comment
-----  -------------------------------- ------- ---- ---- --------------
 1744  577294575*2^481273+1              144887  g42 2008 
 3706  577294575*2^372840+1              112245  g42 2006 
 9756  577294575*2^300263+1               90397  g42 2001 
10390  577294575*2^279263+1               84076  g42 2001 
11886  577294575*2^238106+1               71686  g42 2001 
11922  577294575*2^237384+1               71469  g42 2001 
11987  577294575*2^236432+1               71182  g42 2001 
12661  577294575*2^223519+1               67295  g42 2001 
13591  577294575*2^207060+1               62341  g42 2000

and another 10. Search the Top-5000 site for more.

[roger]

Yeah, I just found the top-5000 listing for g42, though it only showed the top two primes you listed.

I really thought no one had been doing this, no one seemed to be doing larger k-values except Robert44444uk here.

This has been my major proth prime search for quite a while, and I found 139 primes, along with the last 5 that Kosmaj listed.

I assume g42 has all the lower primes as well? And there's no gaps?

[VBCurtis]

We have no reason to know, but if you found the exact primes he did, you should assume he is searching in order. We'd know more about him/her than "g42" if we had any contact, so we're just guessing.
-Curtis

[Happy5214]

Actually, he has his e-mail posted on his bio page. It's jack at brennen dot net. You might want to try contacting him there.

-Happy

[Alex]

Hello!

577294575*2^n+1 is a good candidate for Prime Wiki Category: Proth 2 Count-100.

There are only 4 k`s now:
k = 555 (101 known primes)
k = 2441205 (106 known primes)
k = 30562993973479965532402725 (180 known primes)
k = 32268186233370440249391495 (177 known primes)

In 2021 I have noticed k = 2441205 with 96 primes and no Max n reported.
I have checked it up to n = 500,000 and have found 10 new primes.
Then k = 2441205 became the Category Count-100 member :-)

Roger, could you create such a page or just post your results here?

P.S. The History of Count-100 began with Puzzle 6. Ray Ballinger suggestion
(There is another Category candidate k = 945561887392230553579269135)

[kar_bon]

This is an ambitious wish to a user after 13 years to create a Wiki-page or posts results here.

I've done this page including small primes n<50k and twins, also added all Top5000 known primes.
I don't know the highest n-limit of the search and if any prime upto the lastest is missing.
So 144 primes known so far for this Proth-k.

[Alex]

Originally Posted by kar_bon

...

Very well!
That is a happy ending!
I have found Brennen`s early result 113 / 41312 here (page 362)

I`m sieving for k = 945561887392230553579269135 now (and I`ll create such a page soon).

P.S. Many other S-candidates were reported here (Very Prime Riesel and Sierpinski k), not sure I`ll take them on.

[Alex]

Hello.
I`ve double-checked 945561887392230553579269135*2^n+1 with n=[1..110.000] (32.987 tests).

142 primes confirmed:

945561887392230553579269135*2^5+1
945561887392230553579269135*2^8+1
945561887392230553579269135*2^11+1
945561887392230553579269135*2^18+1
945561887392230553579269135*2^21+1 (Term1 Cunningham chains 2nd kind)
945561887392230553579269135*2^22+1 (Term2)
945561887392230553579269135*2^23+1 (Term3)
945561887392230553579269135*2^29+1
945561887392230553579269135*2^34+1
945561887392230553579269135*2^42+1
945561887392230553579269135*2^57+1 (Term1 Cunningham chains 2nd kind)
945561887392230553579269135*2^58+1 (Term2)
945561887392230553579269135*2^64+1
945561887392230553579269135*2^68+1
945561887392230553579269135*2^73+1 (Term1 Cunningham chains 2nd kind)
945561887392230553579269135*2^74+1 (Term2)
945561887392230553579269135*2^77+1
945561887392230553579269135*2^94+1 (Term1 Cunningham chains 2nd kind)
945561887392230553579269135*2^95+1 (Term2)
945561887392230553579269135*2^102+1 (Term1 Cunningham chains 2nd kind)
945561887392230553579269135*2^103+1 (Term2)
945561887392230553579269135*2^106+1
945561887392230553579269135*2^109+1
945561887392230553579269135*2^139+1
945561887392230553579269135*2^179+1
945561887392230553579269135*2^197+1
945561887392230553579269135*2^226+1
945561887392230553579269135*2^240+1
945561887392230553579269135*2^247+1
945561887392230553579269135*2^256+1
945561887392230553579269135*2^317+1
945561887392230553579269135*2^329+1
945561887392230553579269135*2^335+1 (Term1 Cunningham chains 2nd kind)
945561887392230553579269135*2^336+1 (Term2)
945561887392230553579269135*2^342+1
945561887392230553579269135*2^386+1
945561887392230553579269135*2^390+1
945561887392230553579269135*2^450+1
945561887392230553579269135*2^455+1
945561887392230553579269135*2^463+1
945561887392230553579269135*2^530+1
945561887392230553579269135*2^574+1
945561887392230553579269135*2^612+1
945561887392230553579269135*2^622+1 (Term1 Cunningham chains 2nd kind)
945561887392230553579269135*2^623+1 (Term2)
945561887392230553579269135*2^700+1
945561887392230553579269135*2^712+1
945561887392230553579269135*2^744+1
945561887392230553579269135*2^867+1
945561887392230553579269135*2^880+1
945561887392230553579269135*2^897+1
945561887392230553579269135*2^906+1
945561887392230553579269135*2^947+1 (Term1 Cunningham chains 2nd kind)
945561887392230553579269135*2^948+1 (Term2)
945561887392230553579269135*2^964+1
945561887392230553579269135*2^967+1
945561887392230553579269135*2^973+1
945561887392230553579269135*2^992+1
945561887392230553579269135*2^1019+1
945561887392230553579269135*2^1035+1
945561887392230553579269135*2^1189+1
945561887392230553579269135*2^1318+1
945561887392230553579269135*2^1337+1
945561887392230553579269135*2^1392+1
945561887392230553579269135*2^1428+1
945561887392230553579269135*2^1478+1
945561887392230553579269135*2^1503+1
945561887392230553579269135*2^1702+1
945561887392230553579269135*2^1710+1 (Twin)
945561887392230553579269135*2^1730+1
945561887392230553579269135*2^1734+1
945561887392230553579269135*2^1792+1
945561887392230553579269135*2^2070+1
945561887392230553579269135*2^2115+1
945561887392230553579269135*2^2374+1
945561887392230553579269135*2^2404+1
945561887392230553579269135*2^2498+1
945561887392230553579269135*2^2810+1
945561887392230553579269135*2^2832+1 (Term1 Cunningham chains 2nd kind)
945561887392230553579269135*2^2833+1 (Term2)
945561887392230553579269135*2^2942+1
945561887392230553579269135*2^3147+1
945561887392230553579269135*2^3386+1
945561887392230553579269135*2^3443+1
945561887392230553579269135*2^3503+1
945561887392230553579269135*2^3603+1
945561887392230553579269135*2^3606+1
945561887392230553579269135*2^3703+1
945561887392230553579269135*2^4065+1
945561887392230553579269135*2^4154+1
945561887392230553579269135*2^4328+1
945561887392230553579269135*2^4778+1
945561887392230553579269135*2^4783+1
945561887392230553579269135*2^5079+1
945561887392230553579269135*2^5294+1
945561887392230553579269135*2^5318+1
945561887392230553579269135*2^5910+1
945561887392230553579269135*2^6276+1
945561887392230553579269135*2^6295+1
945561887392230553579269135*2^6356+1
945561887392230553579269135*2^7725+1
945561887392230553579269135*2^8755+1
945561887392230553579269135*2^8888+1
945561887392230553579269135*2^10347+1
945561887392230553579269135*2^10805+1
945561887392230553579269135*2^10951+1
945561887392230553579269135*2^11016+1
945561887392230553579269135*2^11409+1
945561887392230553579269135*2^11544+1
945561887392230553579269135*2^11806+1
945561887392230553579269135*2^12286+1
945561887392230553579269135*2^13720+1
945561887392230553579269135*2^14253+1
945561887392230553579269135*2^18106+1
945561887392230553579269135*2^18374+1
945561887392230553579269135*2^19209+1
945561887392230553579269135*2^19403+1
945561887392230553579269135*2^19610+1
945561887392230553579269135*2^22867+1
945561887392230553579269135*2^23690+1
945561887392230553579269135*2^24122+1
945561887392230553579269135*2^24764+1
945561887392230553579269135*2^26176+1
945561887392230553579269135*2^30228+1
945561887392230553579269135*2^31284+1
945561887392230553579269135*2^31596+1
945561887392230553579269135*2^31638+1
945561887392230553579269135*2^36412+1
945561887392230553579269135*2^38278+1
945561887392230553579269135*2^51268+1
945561887392230553579269135*2^54207+1
945561887392230553579269135*2^54507+1
945561887392230553579269135*2^57080+1
945561887392230553579269135*2^60436+1
945561887392230553579269135*2^73141+1
945561887392230553579269135*2^74320+1
945561887392230553579269135*2^85981+1
945561887392230553579269135*2^87805+1
945561887392230553579269135*2^103597+1
945561887392230553579269135*2^106347+1
945561887392230553579269135*2^107506+1
945561887392230553579269135*2^109667+1

Prime Wiki page created.

I reserve this for more processing.

[Alex]

Hello.
I`ve checked 945561887392230553579269135*2^n+1 with n=[110.000..340.000] (49.285 tests).

19 new primes found:

945561887392230553579269135*2^110287+1
945561887392230553579269135*2^112283+1
945561887392230553579269135*2^126548+1
945561887392230553579269135*2^129569+1
945561887392230553579269135*2^147357+1
945561887392230553579269135*2^154440+1
945561887392230553579269135*2^159686+1
945561887392230553579269135*2^166271+1
945561887392230553579269135*2^196240+1
945561887392230553579269135*2^197019+1
945561887392230553579269135*2^197804+1
945561887392230553579269135*2^208977+1
945561887392230553579269135*2^224537+1
945561887392230553579269135*2^237930+1
945561887392230553579269135*2^239006+1
945561887392230553579269135*2^246326+1
945561887392230553579269135*2^319664+1
945561887392230553579269135*2^324124+1
945561887392230553579269135*2^333374+1

Now [1..340.000] is a minimal common range for all 6 pages in Category: Proth 2 Count-100:

S 555 : 77 / 340000
S 2441205 : 101 / 340000
S 577294575 : ~141 / 340000 (missing-range)
S 30562993973479965532402725 : 180 / 340000
S 32268186233370440249391495 : 177 / 340000
S 945561887392230553579269135 : 161 / 340000


P.S. I`ll continue processing 945561887392230553579269135:

1 more prime found:

945561887392230553579269135*2^347097+1

S 945561887392230553579269135 : ~162 / 350000 (not fully checked)