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Message boards : General discussion : Untaken types of prime?

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mackerel
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Message 85554 - Posted: 12 May 2015 | 22:57:37 UTC

Setting up the factorial prime challenge made me think again about the formats different types of prime numbers that are around. Most of the ones I've looked at are of the form kb^n+/-1 so I was wondering are there any simple formats that haven't been taken yet?

I thought, wouldn't it be interesting to look for primes in more random things, like the digits of pi? A quick search and sure enough, someone thought of that already. Similarly for e.

http://mathworld.wolfram.com/Pi-Prime.html
http://mathworld.wolfram.com/e-Prime.html

What about putting them together? I propose, pie primes :) That is, multiply pi and e to give 8.53973422267357..., and look for prime numbers in that.

The first pie prime is... 853. Unfortunately I only have Excel so can go to 15 digits, and I'm not sure if the last one is rounded or not so it might not count anyway. No more primes up to that point.

If I get really bored I might look up other constants and start putting them together for more interesting words to look for primes in :)

JeppeSN

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Message 85575 - Posted: 13 May 2015 | 15:16:08 UTC - in response to Message 85554.

There is no quick way to find huge primes in a sequence like that.

I verified that 853 and 8539 are primes, and there are no other primes (not even probable primes) below 10^11200 in this sequence.

/JeppeSN

mackerel
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Message 85580 - Posted: 13 May 2015 | 16:29:41 UTC - in response to Message 85575.

How did I miss 8539? Must have made a typing error when checking manually... agree there is no short cut to finding primes this way, but it was just a little diversion to try and come up with a "new" prime type.

I take it you have software or other method to get a longer expansion of the digits in this case?

serge

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Message 85583 - Posted: 13 May 2015 | 17:18:20 UTC - in response to Message 85580.
Last modified: 13 May 2015 | 17:19:09 UTC

Pari/GP.
Download, install; then use something like this in its command-line:

? \p 15000 realprecision = 15008 significant digits (15000 digits displayed) ? Pi 3.141592653589793238462643383279502884197169399........3710311508984279928 ? write("Pi13999",floor(Pi*10^13999)) ? Pi*exp(1) 8.539734222673567065.............94887320195175 ? write("PiE13999",floor(Pi*exp(1)*10^13999))

Now, that's not the problem. The problem is that when you will find a long enough prime in this decimal extension, -- you will not be able to prove it. Anything above, say, 40,000 digits is a no fly-zone. Practically, you will die of boredom while waiting for a 20-thread Primo to prove even a 20,000-digits prime.

mackerel
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Message 85587 - Posted: 13 May 2015 | 18:38:18 UTC - in response to Message 85583.

I assume after a point, brute force doesn't cut it? And for arbitrary testing of numbers, without special format, there are limited options to perform in a reasonable amount of time?

That software will be fun to play with regardless, and all this is a nice learning exercise for me.

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Message 85594 - Posted: 14 May 2015 | 4:41:03 UTC - in response to Message 85587.

I assume after a point, brute force doesn't cut it? And for arbitrary testing of numbers, without special format, there are limited options to perform in a reasonable amount of time?

Correct. There were invented some special "shortcut" tests which could test numbers in given form (e.g. Proth test for k*b^n+1), but brute-forcing big random number is impossible (btw lot of modern cryptography is also based on this fact).

Message boards : General discussion : Untaken types of prime?