a difficult problem

[enzocreti]

Let be N a nonnegative integer.


I search integers N such that


N+20=s^2*p and N+19=q^2*(2*p+1), where s and q are integers>1 and p is a prime>2 and (2p+1) is a prime as well.




Are there infinitely many such numbers?

[mathwiz]

Originally Posted by enzocreti

Let be N a nonnegative integer.


I search integers N such that


N+20=s^2*p and N+19=q^2*(2*p+1), where s and q are integers>1 and p is a prime>2 and (2p+1) is a prime as well.




Are there infinitely many such numbers?

Why is this relevant to anything? What is the significance of these numbers?

[Uncwilly]

Originally Posted by mathwiz

Why is this relevant to anything? What is the significance of these numbers?

We ask ourselves this all the time about posts here.

[Viliam Furik]

First, you should include examples of the ones you found.

If you found one, two, or any other relatively small amount under reasonable upper bound, and the extending of the bound doesn't seem to help, there is a chance there are only finitely many, and also there is a chance you have found all of them.

It is similar to Fermat primes (2^2^n + 1). There are only 5 of them, for n=0,1,2,3,4. No prime has been found with n > 4. However, the lowest n for which the status is unknown is n=33 (no factor found, and Pepin test not runnable for at least a few years from now).

[LaurV]

Originally Posted by enzocreti

Are there infinitely many such numbers?

Yes. (Un)Surprisingly, they are not rare either. The smallest is 25, and the largest you can find in just few minutes is 4070554079227670608. Your homework is to find those in between.

Edit: Of course, ignoring the fact that the relevance of this, as other posters said, is zero divided by four, and ignoring the fact that the question is somehow idiotically asked (why did you put the 19 and 20 inside? why not 13 and 107, for example? )

[retina]

Originally Posted by Viliam Furik

It is similar to Fermat primes (2^2^n + 1). ... the lowest n for which the status is unknown is n=33 (no factor found, and Pepin test not runnable for at least a few years from now).

You can run the Pepin test now. The problem is just patience, and the willingness of your offspring to continue the test after your passing. The test can be transferred to newer hardware to continue the run. But make sure you use reliable kit, with ECC memory at a minimum, and also some sort of error correction for the save files. And make regular backups.

[Viliam Furik]

Originally Posted by retina

You can run the Pepin test now. The problem is just patience, and the willingness of your offspring to continue the test after your passing. The test can be transferred to newer hardware to continue the run. But make sure you use reliable kit, with ECC memory at a minimum, and also some sort of error correction for the save files. And make regular backups.

I know. By runnable I meant "runnable in a reasonable time", with reasonable being at most 10 years.

---
EDIT:
My calculations, based on the scaling method for Mersenne exponents, tell me it would take at least 3000 years to run Pépin's test for F33 on Radeon VII.

If I wanted to run it right now, I would have to be sure my computation will be continued for a loooooong time after not only I die, but also my offsprings, and their offsprings, and so on. There is a very high chance that, within those 3000 years, at least one of the following events will happen:
- All (or most) of humanity dies because of a plague or a big war.
- Some big chunk of rock strikes the planet.
- The binary computing becomes obsolete, because of quantum computers or other possible breakthroughs, resulting in my computation becoming impossible to continue.
- Computational capabilities become so big that my test can be run under a year.
- Someone discovers a way to check Mersenne and Fermat primes instantly.

I think anyone will agree, that at least one of these events happening has a reasonably high chance for me to not start the test at all.

[a1call]

Originally Posted by Viliam Furik

- Someone discovers a way to check Mersenne and Fermat primes instantly.

Now that would be a very hard pill to swallow around here.

[axn]

Originally Posted by a1call

Now that would be a very hard pill to swallow around here.

On the contrary, that would be a very welcome development. We don't suffer from sunk cost fallacy here.

[a1call]

Well it is hypothetical and very unlikely to happen. But I think it would be very difficult to fill the resulting void of purpose and activity on this board. It would render all the distributed computing and record keeping null.
You are welcome to disagree, but I think very often the voyage is more rewarding than the destination. It is more fun to look for primes when it's challenging than when it's child's play.

[retina]

Originally Posted by a1call

It is more fun to look for primes when it's challenging than when it's child's play.

I doubt that would happen. Everything has limitations. All that would happen is the wavefront moves ahead a few orders of magnitude. So instead of searching at 100M we then search at 100T (or whatever, 100P, ...). At the scale if infinity "instant" is a very long time.