Continuity of Primes

[LiquidNitrogen]

With all of the various ways to generate primes (Riesel, Proth, Mersenne etc) there seems to be quite a few "gaps" that must occur in the "plain old" primes that can't be expressed with an elegant, concise formula that lends itself to fast primality proving.

Is there any such list of "known continuous primes" where all the primes < P have been identified?

That would be the real "No Primes Left Behind" effort I would think.

[Mr. P-1]

http://primes.utm.edu/notes/faq/LongestList.html

[henryzz]

Primegrid had a subproject that found contiuous primes. They ended up with with many dvds of data.

[ATH]

Here you can find the first 10¹² primes up to 3*10¹³:
http://primes.utm.edu/nthprime/

[LiquidNitrogen]

Originally Posted by Mr. P-1

http://primes.utm.edu/notes/faq/LongestList.html

So I guess that means when we "discover" primes of the various forms, they are much larger than the primes that are known to exist in continuity, and therefore they are "real discoveries," in a manner of speaking.

If that is true, each prime is probably a unique find.

Correct?

[Mr. P-1]

Originally Posted by LiquidNitrogen

So I guess that means when we "discover" primes of the various forms, they are much larger than the primes that are known to exist in continuity, and therefore they are "real discoveries," in a manner of speaking.

According to the article: "At the time I last updated this page, these projects had found (but not stored) all the prime up to 10^18, but not yet to 10^19."

If you expended 100 times as much effort, you might get up to 10^21. If you devoted the entire world's computer resources to the project, you could probably push it well past 10^30.

You'd never, ever, reach this 100 digit prime:

3664461208681099176204078925954510073897620111029087350504719136242910190767917650858670935504633223

If that is true, each prime is probably a unique find.

I don't know what you mean by "unique" in this context. Here's the next one:

3664461208681099176204078925954510073897620111029087350504719136242910190767917650858670935504633509

Both took a fraction of a second to generate on my computer. Neither, in all probability, has ever been "discovered" before.

The primes that are considered "discoveries" are the ones that take significant resources to find

I suggest you read this primer on primality testing. You'll have a much better understanding of what you see in this forum.

[LiquidNitrogen]

Originally Posted by Mr. P-1

If you expended 100 times as much effort, you might get up to 10^21. If you devoted the entire world's computer resources to the project, you could probably push it well past 10^30.

You'd never, ever, reach this 100 digit prime:

That makes the point crystal clear. I haven't seen any online resources offering such a concise explanation.

Originally Posted by Mr. P-1

I don't know what you mean by "unique" in this context.

The first answer you gave answers this one. With such a huge gap in the prime record, there is no way any of the primes we generate here are a part of that continuous list. I thought maybe there were people somewhere who would test the neighborhood of announced primes for primality as well, perhaps "finding" some that might have been shown later.

Now I see that was a stupid assumption!

So each prime that is found is, essentially, a new find. I'd call that a discovery. The large primes you mentioned I would call a "monumental undertaking" as well.

Originally Posted by Mr. P-1

I suggest you read this primer on primality testing. You'll have a much better understanding of what you see in this forum.

Primer on primes. Nice!

[LiquidNitrogen]

Originally Posted by Mr. P-1

I suggest you read this primer on primality testing.

I did find one typo on http://primes.utm.edu/prove/prove1.html

In 2002 a long standing question was answered: can integers be prove prime

I think this should be changed to the word proven there.

And now I actually know how to do the Lucas-Lehmer test, although those s(k) numbers grow too big for Excel after s(4). At least Excel can prove 2^5 - 1 is prime using Lucas-Lehmer

[CRGreathouse]

Originally Posted by LiquidNitrogen

And now I actually know how to do the Lucas-Lehmer test, although those s(k) numbers grow too big for Excel after s(4). At least Excel can prove 2^5 - 1 is prime using Lucas-Lehmer

If you reduce mod the Mersenne number at each step, you can prove 2^19 - 1 prime in Excel.

[LiquidNitrogen]

Originally Posted by CRGreathouse

If you reduce mod the Mersenne number at each step, you can prove 2^19 - 1 prime in Excel.

I'm not sure I follow. Here is what I did.

1. Proving p = 2^n - 1 is prime for n = 5, p = 31.
2. S(0) = 4 {defined}
3. Need to generate up to S(n-2) where S(x+1) = [S(x) * S(x)] - 2

3a. S(1) = 4^2 - 2 = 14
3b. S(2) = 14^2 - 2 = 194
3c. S(3) = 194^2 - 2 = 37634

4. Test S(n-2)/p = S(3)/p = 37634/31. If remainder is 0, p is prime.

37634/31 = 1214.0 so p is prime.

What would this involve doing it the way you mentioned?

[retina]

Originally Posted by LiquidNitrogen

I'm not sure I follow. Here is what I did.

1. Proving p = 2^n - 1 is prime for n = 5, p = 31.
2. S(0) = 4 {defined}
3. Need to generate up to S(n-2) where S(x+1) = [S(x) * S(x)] - 2

3a. S(1) = 4^2 - 2 = 14
3b. S(2) = 14^2 - 2 = 194
3c. S(3) = 194^2 - 2 = 37634

4. Test S(n-2)/p = S(3)/p = 37634/31. If remainder is 0, p is prime.

37634/31 = 1214.0 so p is prime.

What would this involve doing it the way you mentioned?

3. Need to generate up to S(n-2) where S(x+1) = {[S(x) * S(x)] - 2} mod p