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Message boards : AP26 - AP27 Search : AP Prime Numbers

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Michael Becker

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Message 100730 - Posted: 8 Nov 2016 | 19:11:15 UTC

Hi,

i'm interested if prime numbers of AP findings are worth to be reported, and if they can displayed in the 'Top Prime Finders' table.
I'm also interested to know how long they are (how many digits in decimal), and also how big is the gap between the primes in a row (maybe i can calculate myself).

Michael

Michael Goetz
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Message 100734 - Posted: 8 Nov 2016 | 21:00:18 UTC - in response to Message 100730.

Hi,

i'm interested if prime numbers of AP findings are worth to be reported, and if they can displayed in the 'Top Prime Finders' table.
I'm also interested to know how long they are (how many digits in decimal), and also how big is the gap between the primes in a row (maybe i can calculate myself).

Michael

The prime numbers themselves are not particularly interesting. They're very small. It's the fact that they form a long sequence of prime numbers that makes them interesting.

For example, the first prime number in the AP26 discovered recently is 149836681069944461. That's 18 digits. That's about 388 thousand digits too small to qualify for the Top 5000 primes.
____________
My lucky number is 75898524288+1

Michael Becker

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Message 100739 - Posted: 8 Nov 2016 | 21:33:58 UTC - in response to Message 100734.

Hi Michael,

thank you for the information.
I didn't thought that they are so small.

Michael

Yair Givoni

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Message 109263 - Posted: 8 Aug 2017 | 6:49:52 UTC
Last modified: 8 Aug 2017 | 6:50:53 UTC

Curiosity, does anybody know what happens primenumberwise in between the primes in the AP series? Does AP series overlap?

JeppeSN

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Message 109315 - Posted: 11 Aug 2017 | 13:03:15 UTC - in response to Message 109263.

Curiosity, does anybody know what happens primenumberwise in between the primes in the AP series? Does AP series overlap?

There will almost always be other primes in between the primes in an AP series.

Finding an instance where that is not the case, is much harder; it is called a CPAP with the terminology used e.g. in http://primerecords.dk/.

For your other question, yes, AP series will overlap in general. For example {31, 37, 43} is an AP, and {29, 41, 53} is an AP as well.

/JeppeSN

composite
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Message 109316 - Posted: 11 Aug 2017 | 13:31:35 UTC - in response to Message 109315.

AP series will overlap in general. For example {31, 37, 43} is an AP, and {29, 41, 53} is an AP as well.

Also, AP series may have primes in common. For example {31, 37, 43} and {19, 31, 43}.

JeppeSN

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Message 109319 - Posted: 11 Aug 2017 | 17:28:00 UTC - in response to Message 109316.

AP series will overlap in general. For example {31, 37, 43} is an AP, and {29, 41, 53} is an AP as well.

Also, AP series may have primes in common. For example {31, 37, 43} and {19, 31, 43}.

And you expect each odd prime p to be a member of infinitely many arithmetic progressions of primes.

However, for a fixed prime p, the length of the APs it can be a member of is bounded. For example, for p=31, no AP of primes (one of which is 31) can have a length of 32 (or more), because then one other member of the AP would be a multiple of 31, and hence composite.

/JeppeSN

Jay

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Message 109320 - Posted: 11 Aug 2017 | 18:09:13 UTC

Are AP series generally interesting mathematical oddities only, or is there an application where they can be put to use in the real world?

JeppeSN

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Message 109325 - Posted: 12 Aug 2017 | 12:01:48 UTC - in response to Message 109320.

Are AP series generally interesting mathematical oddities only, or is there an application where they can be put to use in the real world?

Mathematics is the real world, you know. Everything else is just impure approximations to it.

But I do not think there are any practical applications. The main result (2004) is that they come in all lengths (Greenâ€“Tao, free access to full article The primes contain arbitrarily long arithmetic progressions (2008)).

/JeppeSN

Jay

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Message 109332 - Posted: 12 Aug 2017 | 19:07:06 UTC - in response to Message 109325.

Mathematics is the real world, you know. Everything else is just impure approximations to it.

Never heard it that way before, but I like it.

But I do not think there are any practical applications.

Thanks, for some reason, in my head, I seem to get them crossed with the optimal Golomb rulers that distributed.net is working on.

Message boards : AP26 - AP27 Search : AP Prime Numbers