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Message boards : 321 Prime Search : Amicable numbers

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Bur
Volunteer tester

Joined: 25 Feb 20
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Message 144948 - Posted: 2 Nov 2020 | 8:33:23 UTC

With the new find I did some reading on Thabit/321 primes and read they were originally studied for their relation to amicable numbers. While this is likely known to many here, maybe not everyone knows, so:

If all three of p = 3 * 2^n - 1, q = 3 * 2^(n-1) - 1, and r = 9 * 2^(2n-1) - 1 are prime, then 2^n * p * q and 2^n * r are a pair of amicable numbers.

So far only n = 2, 4, and 7 are known to fulfill the requirements. Are there any ideas as to the density of such numbers n? A very rough estimation of their probability would be (1/ln(p))^3, I guess?
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1281979 * 2^485014 + 1 is prime ... no further hits up to: n = 5,700,000

JeppeSN

Joined: 5 Apr 14
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Message 144949 - Posted: 2 Nov 2020 | 9:50:54 UTC

Interesting topic.

Finding two consecutive lucky exponents n-1 and n in the 321 (c=-1) search alone seems improbable. And in that case, having 9*2^(2n-1) - 1 prime as well, ... it will never happen anymore, I would say.

Note, if 3*2^(n-1) - 1 and 3*2^n - 1 are both prime, then the former is a Sophie Germain prime. Finding a Sophie Germain prime near the leading edge for the 321 project would be crazy. Finding so huge SG primes would be really difficult even if you used a sieve that threw away all candidates where the was a small factor of the neighboring candidates.

It was well done by Thâbit in the 9th century to come up with a pattern that matched some of the smallest amicable pairs, but unfortunately I think the family generated by his formula consists of only the three amicable pairs.

However, I think people believe there are infinitely many amicable pairs that do not come from Thâbit's formula, but this is likely very hard to prove.

/JeppeSN

Message boards : 321 Prime Search : Amicable numbers

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