Thread for posting tiny primes

[3.14159]

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.)

[3.14159]

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.

[3.14159]

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)

[kar_bon]

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!

[3.14159]

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)

[kar_bon]

Originally Posted by 3.14159

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!

[3.14159]

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

[kar_bon]

Originally Posted by 3.14159

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!

[3.14159]

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.

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.

[kar_bon]

Originally Posted by 3.14159

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.

[3.14159]

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.