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Old 2010-07-24, 18:18   #1
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Default Thread for posting tiny primes

If you're willing to post some, post away. Please refrain from posting the decimal expansion of the number you're submitting, and please ensure the following:

1. Has no factors below 230
2. Passes a pseudoprimality test (Recommendation: 1-3 bases)
3. Is not a "small" prime. (Please ensure it is ≥ 1000 digits.)

Submissions by me:
12085 * 26000 + 1 (1811 digits)
895 * 27526 + 1 (2269 digits)
1502048 + 1 (4457 digits)
9731 * 12962600 + 1 (8097 digits)
1219 * 26394 + 1 (1928 digits)
15344096 + 1 (13050 digits)
10462 * 12968192 + 1 (25503 digits)
59991 * 291360 + 1 (27507 digits)
2 * 856! + 1 (2140 digits)
2 * 969! + 1 (2475 digits)

Expected primes:
k * 770968192 + 1 (40075-40080 digits) (To be found tonight or tomorrow.)

Last fiddled with by 3.14159 on 2010-07-24 at 19:18
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Old 2010-07-24, 19:19   #2
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Disregard the last two, they're divisible by 859 and 71833. (Mods, please delete the last two on the list.)

More submissions:
1125 * 26300 + 1 (1900 digits)
39600256 + 1 (1178 digits)
25201024 + 1 (3484 digits)
192 * p124# 5 + 1 (1436 digits)
Still looking for more Proth-GFNs. I figured they would be easy to find in the 10-15k digit range.

Last fiddled with by 3.14159 on 2010-07-24 at 20:05
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Old 2010-07-24, 22:36   #3
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A few are probably well-known cases. (Ex: The generalized Fermat numbers I listed.). Both searches haven't turned up much of anything as of yet (40075-40080 digit prime, and a 14640-14645 digit prime search. Rather close to 114.)

Some more submissions:
9787 * 26030 + 1 (1820 digits)
4713 * 24713 + 1 (1423 digits)
1065 * 26303 + 1 (1901 digits)
1881 * 26327 + 1 (1908 digits)

Last fiddled with by 3.14159 on 2010-07-24 at 23:04
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Old 2010-07-24, 23:11   #4
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Here's a quick shot:

4972*3^16384+1 is prime! (7821 digits)
13506*3^16384+1 is prime! (7822 digits)
43728*3^16384+1 is prime! (7822 digits)
50490*3^16384+1 is prime! (7822 digits)

So you want to collect those small primes? You could do a list of the Sierpinski (Proth) side of primes like I do for the Riesel side.
I got thousands of them listed!

Last fiddled with by kar_bon on 2010-07-24 at 23:26
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Old 2010-07-24, 23:31   #5
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Quote:
So you want to collect those small primes? You could do a list of the Sierpinski (Proth) side of primes like I do for the Riesel side.
I got thousands of them listed!
Even though this is obvious sarcasm, sure thing: You take k * b^n - 1, and I'll take k * b^n + 1. Hey: We might just bump into a twin prime pair!

I just have a few searches to finish. (A 40k-digit search, and a 14640-digit search.)

More submissions: 1036 * 125012 + 1 (5412 digits)
770 * 125002 + 1 (5401 digits)

Last fiddled with by 3.14159 on 2010-07-24 at 23:37
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Old 2010-07-24, 23:38   #6
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Quote:
Originally Posted by 3.14159 View Post
You take k * b^n - 1, and I'll take k * b^n + 1. Hey: We might just bump into a twin prime pair!

I just have a few searches to finish.
Sure could happen, if you'll convert your prime (k*b^n+1) into my 'twin-prime' k*2^n-1!

I only list base-2 primes. Perhaps you can program a converter from k*b^n+1 to k*2^b+1 and I can tell you, if I got the other half of the twin!
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Old 2010-07-24, 23:39   #7
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Quote:
Originally Posted by kar_bon
I only list base-2 primes. Perhaps you can program a converter from k*b^n+1 to k*2^b+1 and I can tell you, if I got the other half of the twin!
You only post primes using base 2? Why not generalize further, and include variable bases?
I expect the 14640-digit search to be finished later today.

The search is k * 37544096 + 1
The main search I'm concerned about: k * 779068192 + 1

Last fiddled with by 3.14159 on 2010-07-24 at 23:42
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Old 2010-07-24, 23:43   #8
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Quote:
Originally Posted by 3.14159 View Post
You only post primes using base 2? Why not generalize further, and include variable bases?
What to generalize? Which bases I choose and display/collect? Which ranges of k-values?

Do you know the amount of work only to collect Riesel-primes? Seems not!
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Old 2010-07-24, 23:45   #9
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Quote:
Originally Posted by kar_bon
What to generalize? Which bases I choose and display/collect? Which ranges of k-values?
The k-values are already variable. The bases would be next to be generalized from there.

Quote:
Originally Posted by kar_bon
Do you know the amount of work only to collect Riesel-primes? Seems not!
You mean, a prime number of the form k * 2n - 1? It should be equally difficult as a Proth search.

Last fiddled with by 3.14159 on 2010-07-24 at 23:47
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Old 2010-07-25, 00:08   #10
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Quote:
Originally Posted by 3.14159 View Post
You mean, a prime number of the form k * 2n - 1? It should be equally difficult as a Proth search.
Oh please, don't edit your posts while others want to response to them!

Your original question: No, Riesel numbers are known and the Riesel problem want to find the smallest of them (k=509203 seems the candidate but not proven yet).

Riesel primes are so called, because H.Riesel listed k*2^n-1 for some small k-values and small n-values first (in the 1950's if I'm right).

And yes: The difficulty for testing Proth or Riesel primes are the same.

Last fiddled with by kar_bon on 2010-07-25 at 00:11
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Old 2010-07-25, 00:15   #11
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Quote:
Originally Posted by kar_bon
And yes: The difficulty for testing Proth or Riesel primes are the same.
Excellent! Now, if only these two searches would yield anything, so I can make more submissions. Probability is not on my side at the moment.

Last fiddled with by 3.14159 on 2010-07-25 at 00:17
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