Generalized Cullen and Woodall Searches

[BudgieJane]

Originally Posted by rogue

With help from grueny, all bases from 101 to 10000 have been searched to to n=10000. I have attached the list for Steven to post on his website.

There are two things to note. First, for bases 101 to 200, this list is merged with what Daniel Hermle had on his website, but since the internet wayback machine does not have some of his pages archived, any primes found on his search for n > 10000 are lost and those ranges need to be redone. He has told me via e-mail that he has that information, but until I get it, I will not include those results. Second, I excluded primes for n=1. Someone is welcome to retest these 9900 bases for n=1 to fill in that gap.

There seems to be an error of omission on the line for base 162:

162 [ 95000] 2, 31, 135, 847, 139, 7255, 34051

That 139 should be 1339.

Hmm. I'll have to take a look and see what I did to mess that up.

[Thomas11]

Originally Posted by rogue

Someone is welcome to retest these 9900 bases for n=1 to fill in that gap.

I did a few updates to your GC list:

• For b=162 the 139 was corrected to 1339.
• The primes for n=1 are added (1203 in total).
• Some other missing primes found by Kosmaj and myself (e.g. for bases 216, 256, 7776, 8192 and a few more) were also added.
• And finally I updated some of the test limits according to our notes and records. Some of this information dates back to communication with Daniel Hermle a few years ago (e.g. some of the progress in the b=101-200 interval is actually his work).

While working with your file I noticed a few cases where the n values are out of order, e.g. for b=3164 and b=3168.
So far I only corrected this for b=3168, which I tested up to n=4000.

[rogue]

Originally Posted by Thomas11

I did a few updates to your GC list:

• For b=162 the 139 was corrected to 1339.
• The primes for n=1 are added (1203 in total).
• Some other missing primes found by Kosmaj and myself (e.g. for bases 216, 256, 7776, 8192 and a few more) were also added.
• And finally I updated some of the test limits according to our notes and records. Some of this information dates back to communication with Daniel Hermle a few years ago (e.g. some of the progress in the b=101-200 interval is actually his work).

While working with your file I noticed a few cases where the n values are out of order, e.g. for b=3164 and b=3168.
So far I only corrected this for b=3168, which I tested up to n=4000.

That is due to how I build the list. The primes were not sorted, so some might be out of sequence.

[Thomas11]

Sorry, I just noticed that I somehow messed it up and added the 1's for the GW primes instead of the GC primes.
So here comes the corrected version of the file.

I also noticed quite a few duplicates in the original file for bases < 200 and n<10. This has been corrected now, but there may be still some other cases that I missed...

[rogue]

I just noticed today that Steven Harvey has posted the list of Generalized Cullens for 100 < b <= 10000. All b < 201 have been searched to n=100000 and all b > 200 have been searched to 10000.

[sweety439]

Why the page of generalized Woodall primes (n*b^n-1) has a condition that n>=b-1, but the page of generalized Cullen primes (n*b^n+1) does not have?

The pages are:

Generalized Woodall primes: http://harvey563.tripod.com/GWlist.txt

Generalized Cullen primes: http://www.loeh.name/guenter/gc/status.html for b<=100, http://harvey563.tripod.com/GClist.txt for b>100.

[rogue]

Read this to find your answer: http://primes.utm.edu/glossary/page....=WoodallNumber

[sweety439]

Originally Posted by rogue

Read this to find your answer: http://primes.utm.edu/glossary/page....=WoodallNumber

This is also true for generalized Cullen primes: http://primes.utm.edu/glossary/xpage/Cullens.html.

[rogue]

The difference is in the definition:

The reason for the restriction on the exponent n is simple, without some restriction every prime p would be a generalized Woodall.

so any prime that can be written in this form could be called a generalized Cullen prime

This means that some p can be generalized Cullens, but all p can be generalized Woodalls.

[sweety439]

Originally Posted by rogue

The difference is in the definition:

The reason for the restriction on the exponent n is simple, without some restriction every prime p would be a generalized Woodall.

so any prime that can be written in this form could be called a generalized Cullen prime

This means that some p can be generalized Cullens, but all p can be generalized Woodalls.

No, let n=1, then every prime p can be written as 1*(p-1)^1+1 (generalized Cullen base p-1) and 1*(p+1)^1-1 (generalized Woodall base p+1).

[science_man_88]

Originally Posted by sweety439

No, let n=1, then every prime p can be written as 1*(p-1)^1+1 (generalized Cullen base p-1) and 1*(p+1)^1-1 (generalized Woodall base p+1).

ah but without base 1, 2 can't be represented that way. admittedly I didn't think of that right away either.