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Message boards : Sophie Germain Prime Search : SGS primes decimal length

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Usucapio LibertatisProject donor
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Joined: 21 Apr 10
Posts: 729
ID: 59072
Credit: 543,201,357
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Discovered 1 mega prime2012 Tour de Primes highest prime scoreFound 2 primes in the 2018 Tour de Primes321 LLR Jade: Earned 10,000,000 credits (14,116,665)Cullen LLR Ruby: Earned 2,000,000 credits (2,622,415)ESP LLR Ruby: Earned 2,000,000 credits (3,461,313)Generalized Cullen/Woodall LLR Amethyst: Earned 1,000,000 credits (1,521,709)PPS LLR Jade: Earned 10,000,000 credits (10,419,603)PSP LLR Ruby: Earned 2,000,000 credits (2,387,311)SoB LLR Ruby: Earned 2,000,000 credits (3,923,258)SR5 LLR Ruby: Earned 2,000,000 credits (2,285,621)SGS LLR Ruby: Earned 2,000,000 credits (2,349,916)TRP LLR Ruby: Earned 2,000,000 credits (2,527,307)Woodall LLR Amethyst: Earned 1,000,000 credits (1,295,630)321 Sieve Amethyst: Earned 1,000,000 credits (1,664,568)Cullen/Woodall Sieve (suspended) Ruby: Earned 2,000,000 credits (4,449,019)Generalized Cullen/Woodall Sieve (suspended) Ruby: Earned 2,000,000 credits (2,388,253)PPS Sieve Double Silver: Earned 200,000,000 credits (337,847,415)Sierpinski (ESP/PSP/SoB) Sieve (suspended) Ruby: Earned 2,000,000 credits (2,072,601)TRP Sieve (suspended) Ruby: Earned 2,000,000 credits (2,441,257)AP 26/27 Sapphire: Earned 20,000,000 credits (21,225,750)GFN Sapphire: Earned 20,000,000 credits (44,249,129)PSA Emerald: Earned 50,000,000 credits (79,952,619)
Message 42156 - Posted: 25 Oct 2011 | 13:46:57 UTC

This may be a silly question, but I was wondering why so many (if not all) SGS found primes have exactly 200700 digits. Could there be as many with, say, 200701 or 200699?

How come so many different exponents (and at least four different powers) generate results with the same length in decimal representation?
____________
676754^262144+1 is prime

HonzaProject donor
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Joined: 15 Aug 05
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Discovered 5 mega primesEliminated 3 conjecture "k"sFound 2 primes in the 2018 Tour de PrimesFound 1 prime in the 2018 Tour de Primes Mountain Stage2019 Tour de Primes largest primeFound 4 primes in the 2019 Tour de PrimesFound 1 mega prime in the 2019 Tour de PrimesFound 1 prime in the 2019 Tour de Primes Mountain Stage321 LLR Emerald: Earned 50,000,000 credits (53,741,535)Cullen LLR Emerald: Earned 50,000,000 credits (50,296,190)ESP LLR Emerald: Earned 50,000,000 credits (50,853,190)Generalized Cullen/Woodall LLR Emerald: Earned 50,000,000 credits (50,309,119)PPS LLR Double Bronze: Earned 100,000,000 credits (100,184,938)PSP LLR Sapphire: Earned 20,000,000 credits (37,714,448)SoB LLR Emerald: Earned 50,000,000 credits (67,865,553)SR5 LLR Emerald: Earned 50,000,000 credits (50,320,316)SGS LLR Sapphire: Earned 20,000,000 credits (26,720,423)TPS LLR (retired) Bronze: Earned 10,000 credits (43,033)TRP LLR Emerald: Earned 50,000,000 credits (57,441,313)Woodall LLR Emerald: Earned 50,000,000 credits (50,956,094)321 Sieve Emerald: Earned 50,000,000 credits (50,611,470)Cullen/Woodall Sieve (suspended) Ruby: Earned 2,000,000 credits (4,142,109)Generalized Cullen/Woodall Sieve (suspended) Emerald: Earned 50,000,000 credits (50,504,945)PPS Sieve Double Gold: Earned 500,000,000 credits (504,377,255)Sierpinski (ESP/PSP/SoB) Sieve (suspended) Sapphire: Earned 20,000,000 credits (20,288,222)TRP Sieve (suspended) Sapphire: Earned 20,000,000 credits (20,149,354)AP 26/27 Double Silver: Earned 200,000,000 credits (214,118,887)GFN Double Gold: Earned 500,000,000 credits (633,106,043)PSA Double Gold: Earned 500,000,000 credits (531,254,992)
Message 42164 - Posted: 25 Oct 2011 | 18:32:46 UTC

2^30 = 1 073 741 824
2^31 = 2 147 483 648
2^32 = 4 294 967 296
2^33 = 8 589 934 592
All with exactly the same number of digits.
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Badge score: 1*1 + 5*1 + 7*1 + 8*8 + 9*7 + 11*1 + 12*3 = 187

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321 LLR Jade: Earned 10,000,000 credits (10,043,076)Cullen LLR Jade: Earned 10,000,000 credits (10,256,286)ESP LLR Jade: Earned 10,000,000 credits (10,075,118)Generalized Cullen/Woodall LLR Jade: Earned 10,000,000 credits (10,184,133)PPS LLR Emerald: Earned 50,000,000 credits (54,952,284)PSP LLR Jade: Earned 10,000,000 credits (10,037,823)SoB LLR Jade: Earned 10,000,000 credits (10,053,795)SR5 LLR Sapphire: Earned 20,000,000 credits (20,710,959)SGS LLR Jade: Earned 10,000,000 credits (10,026,927)TRP LLR Jade: Earned 10,000,000 credits (14,204,693)Woodall LLR Jade: Earned 10,000,000 credits (18,654,724)321 Sieve Silver: Earned 100,000 credits (200,220)Cullen/Woodall Sieve (suspended) Turquoise: Earned 5,000,000 credits (6,710,149)Generalized Cullen/Woodall Sieve (suspended) Jade: Earned 10,000,000 credits (10,061,184)PPS Sieve Emerald: Earned 50,000,000 credits (57,182,910)Sierpinski (ESP/PSP/SoB) Sieve (suspended) Ruby: Earned 2,000,000 credits (2,251,279)TRP Sieve (suspended) Ruby: Earned 2,000,000 credits (3,572,416)AP 26/27 Gold: Earned 500,000 credits (543,507)GFN Sapphire: Earned 20,000,000 credits (21,199,917)PSA Double Silver: Earned 200,000,000 credits (248,258,813)
Message 42168 - Posted: 25 Oct 2011 | 20:53:27 UTC - in response to Message 42164.

Ok, then here is a question that TheDawgz have been trying to find an answer for for a very long time.
How does one predict the decimal length for a given prime or potential prime?
Or for that matter the number of digits in the result of any calculation in the form of: A*B^C?
For example: Honza's prime from the PRPNet GFN32768 sub-project 1*4108672^32768+1 which TheDawgz know to be 216,718 digits long (we looked it up).
Or in the case of one of TheDawgz primes: 789*2^1114779+1 which TheDawgz know to be 335,585 digits long (we were told).
TheDawgz are fairly sure that the answer is obvious, but our furry little brains just can't see it.
____________
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Konstantin Manz
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321 LLR Silver: Earned 100,000 credits (115,153)Cullen LLR Bronze: Earned 10,000 credits (10,974)ESP LLR Bronze: Earned 10,000 credits (11,571)PPS LLR Ruby: Earned 2,000,000 credits (2,074,773)PSP LLR Silver: Earned 100,000 credits (104,595)SoB LLR Silver: Earned 100,000 credits (126,157)SR5 LLR Silver: Earned 100,000 credits (149,712)SGS LLR Silver: Earned 100,000 credits (277,818)TRP LLR Gold: Earned 500,000 credits (508,734)Woodall LLR Bronze: Earned 10,000 credits (17,219)PPS Sieve Ruby: Earned 2,000,000 credits (2,119,298)Sierpinski (ESP/PSP/SoB) Sieve (suspended) Bronze: Earned 10,000 credits (10,227)TRP Sieve (suspended) Bronze: Earned 10,000 credits (21,181)GFN Bronze: Earned 10,000 credits (31,562)PSA Jade: Earned 10,000,000 credits (10,617,327)
Message 42170 - Posted: 25 Oct 2011 | 21:11:42 UTC - in response to Message 42168.

You simply take the logarithm with base 10 of the number n.
log[10](n) = log(n) / log(10)
(log is the natural logarithm)

Usucapio LibertatisProject donor
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Joined: 21 Apr 10
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Discovered 1 mega prime2012 Tour de Primes highest prime scoreFound 2 primes in the 2018 Tour de Primes321 LLR Jade: Earned 10,000,000 credits (14,116,665)Cullen LLR Ruby: Earned 2,000,000 credits (2,622,415)ESP LLR Ruby: Earned 2,000,000 credits (3,461,313)Generalized Cullen/Woodall LLR Amethyst: Earned 1,000,000 credits (1,521,709)PPS LLR Jade: Earned 10,000,000 credits (10,419,603)PSP LLR Ruby: Earned 2,000,000 credits (2,387,311)SoB LLR Ruby: Earned 2,000,000 credits (3,923,258)SR5 LLR Ruby: Earned 2,000,000 credits (2,285,621)SGS LLR Ruby: Earned 2,000,000 credits (2,349,916)TRP LLR Ruby: Earned 2,000,000 credits (2,527,307)Woodall LLR Amethyst: Earned 1,000,000 credits (1,295,630)321 Sieve Amethyst: Earned 1,000,000 credits (1,664,568)Cullen/Woodall Sieve (suspended) Ruby: Earned 2,000,000 credits (4,449,019)Generalized Cullen/Woodall Sieve (suspended) Ruby: Earned 2,000,000 credits (2,388,253)PPS Sieve Double Silver: Earned 200,000,000 credits (337,847,415)Sierpinski (ESP/PSP/SoB) Sieve (suspended) Ruby: Earned 2,000,000 credits (2,072,601)TRP Sieve (suspended) Ruby: Earned 2,000,000 credits (2,441,257)AP 26/27 Sapphire: Earned 20,000,000 credits (21,225,750)GFN Sapphire: Earned 20,000,000 credits (44,249,129)PSA Emerald: Earned 50,000,000 credits (79,952,619)
Message 42171 - Posted: 25 Oct 2011 | 21:26:41 UTC - in response to Message 42168.

Thanks Honza, you've answered my last question. Your example shows that a large number of digits should allow a proportionally large number of results.
The answers to the other questions are also clearer now: the power is much more important to determine the length of a given number than the exponent. If the power is the same or very close to each other as they are on SGS the result will be very close. I think this could help answering the TheDawgz question, or at least an approach to it: if you now the length of another number with the same power, your result should not be very different. If you compare two different a*b^c the one with the largest c will produce a result with the highest number of digits.

I now this must be trivial to those who deal with numbers and number theory. I'm not one of them. Just crunching for fun and thinking out loud right now...
____________
676754^262144+1 is prime

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321 LLR Silver: Earned 100,000 credits (117,797)Cullen LLR Bronze: Earned 10,000 credits (10,073)PPS LLR Silver: Earned 100,000 credits (326,520)PSP LLR Bronze: Earned 10,000 credits (39,120)SoB LLR Bronze: Earned 10,000 credits (20,978)SGS LLR Silver: Earned 100,000 credits (108,795)TRP LLR Silver: Earned 100,000 credits (129,747)Woodall LLR Silver: Earned 100,000 credits (102,198)321 Sieve Silver: Earned 100,000 credits (202,744)Cullen/Woodall Sieve (suspended) Bronze: Earned 10,000 credits (20,797)PPS Sieve Amethyst: Earned 1,000,000 credits (1,379,306)TRP Sieve (suspended) Silver: Earned 100,000 credits (112,603)GFN Silver: Earned 100,000 credits (411,602)PSA Silver: Earned 100,000 credits (238,781)
Message 42292 - Posted: 28 Oct 2011 | 16:49:32 UTC
Last modified: 28 Oct 2011 | 16:49:56 UTC

Another hint if you want to do a quick calculation (without calculator) for getting an approximation (of base 2 numbers):
2^10 ~ 10^3 (1024 vs 1000)

Now you can convert e.g.
2^200000 into 10^60000 (divide by 10, multiply by 3)

If you have base 10, the exponent gives you the number of zeros (or, the number of digits).

If you have something like 357 * xxxx then it has 2-3 digits more (as many digits as the number - here 357 - has). The 2/3 depends on whether it just 'flips' to the next digit or not. (I might be inaccurate here - anyway, it's only about few digits so doesn't really count on the large scale).

Coming now to the Sophies, and give an example here (which is currently on the homepage):
26059520697525*2^666666-1

2^666666 ~ 10^200000
+ 14 digits from the multiplier

So you see, I'm a bit off here, but it gives you the right scale quickly.

Greetings!
____________

Message boards : Sophie Germain Prime Search : SGS primes decimal length

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