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Message boards : 321 Prime Search : Amicable numbers

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Profile BurProject donor
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Joined: 25 Feb 20
Posts: 515
ID: 1241833
Credit: 414,166,440
RAC: 41,145
321 LLR Ruby: Earned 2,000,000 credits (2,092,823)Cullen LLR Ruby: Earned 2,000,000 credits (2,315,295)ESP LLR Ruby: Earned 2,000,000 credits (2,151,088)Generalized Cullen/Woodall LLR Ruby: Earned 2,000,000 credits (2,620,968)PPS LLR Jade: Earned 10,000,000 credits (16,324,584)PSP LLR Ruby: Earned 2,000,000 credits (2,064,832)SoB LLR Ruby: Earned 2,000,000 credits (2,434,466)SR5 LLR Ruby: Earned 2,000,000 credits (2,065,004)SGS LLR Ruby: Earned 2,000,000 credits (2,039,663)TRP LLR Ruby: Earned 2,000,000 credits (2,089,856)Woodall LLR Ruby: Earned 2,000,000 credits (2,112,258)321 Sieve (suspended) Ruby: Earned 2,000,000 credits (2,107,153)PPS Sieve Turquoise: Earned 5,000,000 credits (5,096,952)AP 26/27 Turquoise: Earned 5,000,000 credits (5,797,662)GFN Jade: Earned 10,000,000 credits (11,833,806)WW Double Silver: Earned 200,000,000 credits (349,980,000)PSA Amethyst: Earned 1,000,000 credits (1,042,601)
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?
____________
1281979 * 2^485014 + 1 is prime ... no further hits up to: n = 5,700,000

Profile JeppeSNProject donor
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Joined: 5 Apr 14
Posts: 1804
ID: 306875
Credit: 49,060,680
RAC: 14,163
Found 1 prime in the 2020 Tour de Primes321 LLR Gold: Earned 500,000 credits (593,283)Cullen LLR Gold: Earned 500,000 credits (611,298)ESP LLR Silver: Earned 100,000 credits (174,818)Generalized Cullen/Woodall LLR Silver: Earned 100,000 credits (112,799)PPS LLR Jade: Earned 10,000,000 credits (19,023,839)PSP LLR Silver: Earned 100,000 credits (428,457)SoB LLR Silver: Earned 100,000 credits (466,812)SR5 LLR Silver: Earned 100,000 credits (210,142)SGS LLR Silver: Earned 100,000 credits (136,265)TRP LLR Silver: Earned 100,000 credits (476,246)Woodall LLR Silver: Earned 100,000 credits (281,400)321 Sieve (suspended) Silver: Earned 100,000 credits (175,037)PPS Sieve Bronze: Earned 10,000 credits (10,113)AP 26/27 Bronze: Earned 10,000 credits (12,129)GFN Ruby: Earned 2,000,000 credits (4,977,751)WW Jade: Earned 10,000,000 credits (13,756,000)PSA Turquoise: Earned 5,000,000 credits (7,614,290)
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 Th√Ębit 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 Th√Ębit's formula, but this is likely very hard to prove.

/JeppeSN

Message boards : 321 Prime Search : Amicable numbers

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