Submissions:
New personal record: 22147 * 2[sup]256720[/sup] + 1 (77285 digits)
4785 * 2[sup]17500[/sup] + 1 (5272 digits)
54345 * 2[sup]39860[/sup] + 1 (12004 digits)

Submissions:
21158 * 144[sup]2860[/sup] + 1 (6178 digits)

Searching for k * 99792[sup]7560[/sup] + 1

You know what?
The primes we called 'binary primes' already has a (bit complicated) name!

Just found this at the internet:
[URL]http://primes.utm.edu/lists/top_ten/topten.pdf[/URL] - See page 6
[quote]An anti-Yarborough prime can have any number of 1's and 0's only.[/quote]
It saids that the largest known such prime (when that list was made) is: 10^30802+1110111*10^15398+1.

And I found this at Top5000: 10^78942+10111100100111101*10^39463+1

Here is all of them, less than 11 digits:
[code]11
101
10111
101111
1011001
1100101
10010101
10011101
10100011
10101101
10110011
10111001
11000111
11100101
11110111
11111101
100100111
100111001
101001001
101001011
101100011
101101111
101111011
101111111
110010101
110101001
110111011
111000101
111001001
111010111
1000001011
1000010101
1000011011
1000110101
1001000111
1001001011
1001010011
1001110111
1010000011
1010000111
1010001101
1010010011
1010011111
1010100011
1010110001
1010111111
1011000101
1011110011
1100001101
1100010001
1100101111
1101001001
1101010111
1101110011
1110011101
1110110011
1111011101
1111100101
1111110001
[/code]

[QUOTE=Merfighters;227401]Just found this at the internet:
[URL]http://primes.utm.edu/lists/top_ten/topten.pdf[/URL] - See page 6
[/QUOTE]

It's totally outdated!

Better use this link of the currently [url=http://primes.utm.edu/top20/index.php]Top 20[/url].

Karsten, I must thank you for reviving the dead threads.

Submissions: (Marked in respective categories they are in.)
#2. 468550 * 1999[sup]5346[/sup] + 1 (17652 digits)
#2. 515361 * 1296[sup]5680[/sup] + 1 (17686 digits)
#2. 63483 * 10[sup]8490[/sup] + 1 (8495 digits)
#2. 570331 * 2[sup]93560[/sup] + 1 (28170 digits)
#2. 7428 * 10[sup]13450[/sup] + 1 (13455 digits)
#2 + #8. 4034 * 1500[sup]4034[/sup] + 1 (12816 digits)
#1. 601969 * 2[sup]78290[/sup] + 1 (23568 digits)
#3. 634 * 2480![sup]2[/sup] + 1 (14690 digits)
#2. 83529 * 48[sup]7890[/sup] + 1 (13270 digits)
#2. 159738 * 756[sup]9570[/sup] + 1 (27553 digits)
#7. 1364 * 4200! + 1 (13399 digits)
#3. 29220 * 6[sup]12860[/sup] + 1 (10012 digits)
#3. 2898 * 1201![sup]4[/sup] + 1 (12720 digits)
#4. 703 * p(125)[sup]66[/sup] + 1 (19104 digits)
#3. 12721 * 9![sup]2560[/sup] + 1 (14238 digits)
#1. 3495 * 2[sup]29680[/sup] + 1 (8939 digits)

More submissions:

#2. 912646 * 798336[sup]20160[/sup] + 1 (118995 digits) + Largest PRP found by me. :smile:

Proving it is a whole different matter. No simple methods here. (Proth's only applies to base 2, if I remember correctly.)

This thread will only buzz with some activity if I keep posting meager-sized primes.

Also: The PRP above is off of ECPP's range by a factor of 5 to 11.

This might be more viable: 13963 * 10[sup]8900[/sup] + 1

[code]13963*10^8900 + 1 may be prime. (a = 2)
13963*10^8900 + 1 is prime! (a = 3) [8905 digits][/code]

[code]13963000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000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is prime.

13963 is also prime!

Find me a larger example!

Primality testing 912646*798336^20160+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 5
Calling Brillhart-Lehmer-Selfridge with factored part 35.70%
912646*798336^20160+1 is prime! (3212.3874s+0.0450s)

[QUOTE=Mathew Steine]912646*798336^20160+1 is prime! (3212.3874s+0.0450s)[/QUOTE]

:party: Confirmed prime!

Also: 50740 * 10[sup]19780[/sup] + 1 is a PRP.

And, what program did you use? LLR?

Oh. WinPFGW.

Checking for false primes.

[quote=3.14159;228131]:party: Confirmed prime!

Also: 50740 * 10[sup]19780[/sup] + 1 is a PRP.

And, what program did you use? LLR?[/quote]
That's PFGW with the -t switch, which does an N-1 primality proof. N-1 is a little slower than a standard PRP test, but can consistently give a full proof of the primality of any number N when N-1 is trivially factorizable (as, for instance, in the case of a k*b^n+1 number). PFGW's -tp switch does an N+1 test, which is the analogue covering k*b^n-1 numbers.

FYI, this is also what Proth.exe uses for primality proofs such as 13963*10^8900+1 that you proved in post #239. It's not quite as fast as PFGW, but works on the same basic idea.

FYI #2: LLR uses a different proof of the N-1/N+1 tests for all non-base-2 k*b^n+-1 numbers (for which the LLR and Proth tests are not applicable). Its method allows it to perform a standard Fermat PRP test (a la PFGW) and then, if it returns PRP, build on the result directly to produce a proof (as opposed to an entirely separate test like PFGW does). Depending on various properties of the exact number in question, it can sometimes finish the proof with comparatively trivial additional calculation on top of the initial PRP; even in the worst case scenario, though (where it effectively has to run an additional PRP in another base in order to make the proof line up), it generally takes no more time than PFGW's total PRP+proof time.

From what I've observed of Proth.exe I have to wonder if it's actually doing the same thing as LLR (that is, a PRP test which is extended to form an N-1/N+1 proof) instead of the separate PRP and proof stages as exemplified by PFGW.

Anyway, to summarize all the above gobbledygook: if you find a PRP of the form k*b^n+-1, the quickest way to prove it is with PFGW's -t (+1) or -tp (-1) switches. Proth.exe will take somewhat longer, though can still work if you're going for the "antique proof program" category. :smile: And LLR can be at least as efficient as PFGW if you want to do the whole search directly with it and have primes tested+proved all in one step.