[QUOTE=3.14159;229240]As if what you give is any more correct.[/QUOTE]

Yes, actually. The value is known [i]unconditionally[/i] to fall in the range

4.34918365 * 10[sup]995[/sup] to
4.34918393 * 10[sup]995[/sup].

which specifically excludes your estimate. Under the Riemann Hypothesis, the value is

[code]4.349183651345391174173494401909821175858835012787749165345668159640929917692017520508025158032672749140951055115782715284329642530237156087125124810263387278508968321658911766258523465886512725977808272634984452430455600066331068026911186986701004309020493441689500534188494303684135964291048326520775148989368729408560690565077811567161853319863692842811424426341225271155850381084777091151663905726140220696441497991610961274270332992598688661934080913700499136923763822695771503812449781520... * 10[sup]995[/sup][/code]

where the next decimal place is almost surely 6 (but this is not known).

Submissions:

For item 1. 13945 * 2[sup]14870[/sup] + 1 (4481 digits)

Verification:
Primality testing 13945*2^14870+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 3
Special modular reduction using all-complex FFT length 1024 on 13945*2^14870+1
Calling Brillhart-Lehmer-Selfridge with factored part 99.91%
13945*2^14870+1 is prime! (0.1941s+0.0007s)

For item 2. 14260 * 18[sup]6870[/sup] + 1 (8628 digits)

Verification:
Primality testing 14260*18^6870+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 13
Special modular reduction using zero-padded FFT length 3072 on 14260*18^6870+1
Calling Brillhart-Lehmer-Selfridge with factored part 75.98%
14260*18^6870+1 is prime! (1.4899s+0.0011s)

Strangely, I can't remember my old record of 8608 digits.. I know I used b = 2 for it, and I think the k value was 28583. (I found this sometime in May or June.)

No such value appeared.. 28583 must have been the exponent.

I used to have it in my old prime collection file, which was wiped away along with the 100 GB I lost.

Probably 1091 * 2^28583 + 1.

Well, there is 1091, 7799, etc..

Wait.. come to think of it.. I think it might have been 1091.

[QUOTE=3.14159;229309]Well, there is 1091, 7799, etc..[/QUOTE]

7799 * 2^28583 + 1 has 8609 digits, so that wasn't your number. There aren't any 8608-digit numbers of the form k * 2^28583 - 1, either.

So, 1091 it is.

Also: Sieving for k * 1999[sup]10480[/sup] + 1.

Base 1999 has given primes somewhat quickly on the occasions that I happened to search for it.

The exponents that I have found a prime for are 5040, 8560, and 5346. All happened quickly.

However, the factorial and primorial bases give primes the fastest.

The current record for item 3 stands at 2093 * 600![sup]26[/sup] + 1, at 36614 digits.

Submission: 41650 * 14[sup]17900[/sup] + 1 (20521 digits)

Verification:


Primality testing 41650*14^17900+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 3
Special modular reduction using all-complex FFT length 6K on 41650*14^17900+1
Calling Brillhart-Lehmer-Selfridge with factored part 73.73%
41650*14^17900+1 is prime! (9.5173s+0.0014s)

Gained another entry for item 2.

Not to mention, it is also in the subset of primes with a prime number of digits.

Submissions: 455679 * 2[sup]89490[/sup] + 1 (26945 digits)

Verification:

Primality testing 455679*2^89490+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 5
Special modular reduction using zero-padded FFT length 10K on 455679*2^89490+1
Calling Brillhart-Lehmer-Selfridge with factored part 99.98%
455679*2^89490+1 is prime! (16.8968s+0.0011s)