[QUOTE=3.14159;232387]I remember a conjecture that reads:

"There is always a prime number between an integer n and its double."
[/QUOTE]

It's proven already for a long time. See [url=http://mathworld.wolfram.com/BertrandsPostulate.html]here[/url] and [url=http://tan.epfl.ch/~rhoades/Notes/bertrandPostulate.pdf]here[/url].

Submissions: 13736*14256^1960+1 (8146 digits)

Verification:

Primality testing 13736*14256^1960+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 5
Special modular reduction using zero-padded FFT length 6K on 13736*14256^1960+1
Calling Brillhart-Lehmer-Selfridge with factored part 45.92%
13736*14256^1960+1 is prime! (3.1291s+0.0009s)

11878*4820^4820+1 (17757 digits)

Verification:

Primality testing 11878*4820^4820+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 3
Special modular reduction using zero-padded FFT length 12K on 11878*4820^4820+1
Calling Brillhart-Lehmer-Selfridge with factored part 64.66%
11878*4820^4820+1 is prime! (15.1218s+0.0013s)

Submissions: 157021*2^48960+1 (14744 digits)

Verification:

Primality testing 157021*2^48960+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 3
Special modular reduction using all-complex FFT length 4K on 157021*2^48960+1
Calling Brillhart-Lehmer-Selfridge with factored part 99.97%
157021*2^48960+1 is prime! (3.2654s+0.0006s)

I found something interesting about odd numbers:

n^(10^n) = 1 mod 10^n, where n is odd and where n =/= 5.

Ex: 3^10000000 = 1 mod 10000000.

And, if this is true, 7^(10^3200) = 1 mod (10^3200)

P.S: 7^(10^3200) ≈ 10^(8.45098040014256830712216258592 * 10^3199)

Also; Since you thought you had found a divisor of (10^10^200) + 1, try 10^(10^6400) + 1.

.. And, I think the divisor axn posted was incorrect. I asked for a factor of 10^10^200 + 1.

He provided the smallest factor of 10^[B]2^200[/B] + 1, which is 267 * 2^202 + 1.

So.. I revoke the newer challenge..

And my original one stands, and there are still no known factors for 10[sup]10[sup]200[/sup][/sup] + 1.

Nevermind.. It has an algebraic factor of 10^2^200+1

Are there any Wilson primes > 563? (Prime such that p^2 divides (p-1)! + 1.)

It's all pretty trivial with a multiplier k;

83^2 | 1661 * 82! + 1.

Another small Generalized Fermat just popped out of my ad hoc search:
4 · 83[SUP]236470[/SUP]+1 :w00t:

Goddammit. No one will be able to compete with that.

453805 digits.

Prove that it is prime.

[QUOTE=3.14159;232525]Prove that it is prime.[/QUOTE]

Watch [URL="http://primes.utm.edu/primes/page.php?id=95243"]this[/URL] space.

Submissions: 13241*396^3585+1 (9317 digits)

Primality testing 13241*396^3585+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 5
Special modular reduction using zero-padded FFT length 5K on 13241*396^3585+1
Calling Brillhart-Lehmer-Selfridge with factored part 40.07%
13241*396^3585+1 is prime! (3.3585s+0.0026s)

Submissions: 573382*1999^11700+1 (38626 digits)

Verification:

Primality testing 573382*1999^11700+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 5
Special modular reduction using zero-padded FFT length 24K on 573382*1999^11700+1
Calling Brillhart-Lehmer-Selfridge with factored part 99.99%
573382*1999^11700+1 is prime! (64.2295s+0.0029s)

30842*396^6585+1 (17111 digits)

Verification:

Primality testing 30842*396^6585+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 5
Special modular reduction using zero-padded FFT length 8K on 30842*396^6585+1
Running N-1 test using base 13
Special modular reduction using zero-padded FFT length 8K on 30842*396^6585+1
Calling Brillhart-Lehmer-Selfridge with factored part 40.08%
30842*396^6585+1 is prime! (16.9682s+0.0010s)

Both entries for item 2.

Something that hasn't been done in a long time;

Submission for item 20: 35523*396^6585 + 1 (17111 digits) (Proved using Proth.exe)

35523*396^6585 + 1 may be prime. (a = 2)
35523*396^6585 + 1 is prime! (a = 7) [17111 digits]