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

 Subscribe SortOldest firstNewest firstHighest rated posts first
Author Message
Usucapio Libertatis

Joined: 21 Apr 10
Posts: 742
ID: 59072
Credit: 638,170,037
RAC: 440,682

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

Honza
Volunteer moderator
Volunteer tester
Project scientist

Joined: 15 Aug 05
Posts: 1902
ID: 352
Credit: 3,555,341,029
RAC: 5,248,369

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.
____________
My stats
Badge score: 1*1 + 5*1 + 8*3 + 9*11 + 10*1 + 11*1 + 12*3 = 186

TheDawgz

Joined: 7 Jul 08
Posts: 2235
ID: 25193
Credit: 587,360,666
RAC: 101,555

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.
____________
There's someone in our head but it's not us.

Omega

Joined: 20 Apr 08
Posts: 181
ID: 21694
Credit: 27,525,942
RAC: 16,195

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 Libertatis

Joined: 21 Apr 10
Posts: 742
ID: 59072
Credit: 638,170,037
RAC: 440,682

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

flat

Joined: 26 Sep 10
Posts: 14
ID: 68738
Credit: 3,239,912
RAC: 0

Message 42292 - Posted: 28 Oct 2011 | 16:49:32 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