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-   -   Thread for posting tiny primes (https://www.mersenneforum.org/showthread.php?t=13650)

3.14159 2010-07-24 18:18

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 2[sup]30[/sup]
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 * 2[SUP]6000[/SUP] + 1 (1811 digits)
895 * 2[SUP]7526[/SUP] + 1 (2269 digits)
150[SUP]2048[/SUP] + 1 (4457 digits)
9731 * 1296[SUP]2600[/SUP] + 1 (8097 digits)
1219 * 2[SUP]6394[/SUP] + 1 (1928 digits)
1534[SUP]4096[/SUP] + 1 (13050 digits)
10462 * 1296[SUP]8192[/SUP] + 1 (25503 digits)
59991 * 2[sup]91360[/sup] + 1 (27507 digits)
2 * 856! + 1 (2140 digits)
2 * 969! + 1 (2475 digits)

Expected primes:
k * 77096[sup]8192[/sup] + 1 (40075-40080 digits) (To be found tonight or tomorrow.)

3.14159 2010-07-24 19:19

Disregard the last two, they're divisible by 859 and 71833. (Mods, please delete the last two on the list.)

More submissions:
1125 * 2[sup]6300[/sup] + 1 (1900 digits)
39600[sup]256[/sup] + 1 (1178 digits)
2520[sup]1024[/sup] + 1 (3484 digits)
192 * p[sub]124[/sub]# [sup]5[/sup] + 1 (1436 digits)
Still looking for more Proth-GFNs. I figured they would be easy to find in the 10-15k digit range.

3.14159 2010-07-24 22:36

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 11[sup]4[/sup].)

Some more submissions:
9787 * 2[SUP]6030[/SUP] + 1 (1820 digits)
4713 * 2[SUP]4713[/SUP] + 1 (1423 digits)
1065 * 2[sup]6303[/sup] + 1 (1901 digits)
1881 * 2[sup]6327[/sup] + 1 (1908 digits)

kar_bon 2010-07-24 23:11

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 [url=www.rieselprime.de]Riesel side[/url].
I got thousands of them listed!

3.14159 2010-07-24 23:31

[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![/QUOTE]

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 * 12[sup]5012[/sup] + 1 (5412 digits)
770 * 12[sup]5002[/sup] + 1 (5401 digits)

kar_bon 2010-07-24 23:38

[QUOTE=3.14159;222699]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.[/QUOTE]

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!

3.14159 2010-07-24 23:39

[QUOTE=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!
[/QUOTE]

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 * 3754[sup]4096[/sup] + 1
The main search I'm concerned about: k * 77906[sup]8192[/sup] + 1

kar_bon 2010-07-24 23:43

[QUOTE=3.14159;222701]You only post primes using base 2? Why not generalize further, and include variable bases?[/QUOTE]

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!

3.14159 2010-07-24 23:45

[QUOTE=kar_bon]What to generalize? Which bases I choose and display/collect? Which ranges of k-values?[/QUOTE]

The k-values are already variable. The bases would be next to be generalized from there.

[QUOTE=kar_bon]Do you know the amount of work only to collect Riesel-primes? Seems not![/QUOTE]

You mean, a prime number of the form k * 2[sup]n[/sup] - 1? It should be equally difficult as a Proth search.

kar_bon 2010-07-25 00:08

[QUOTE=3.14159;222703]You mean, a prime number of the form k * 2[sup]n[/sup] - 1? It should be equally difficult as a Proth search.[/QUOTE]

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.

3.14159 2010-07-25 00:15

[QUOTE=kar_bon]And yes: The difficulty for testing Proth or Riesel primes are the same.
[/QUOTE]

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. :no:


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