20210814, 13:35  #1 
"Matthew Anderson"
Dec 2010
Oregon, USA
2·5·89 Posts 
What are the odds of a semiprime
Hi again all,
I have been helping factordb.com for some time now. There is a report factors text box, and I have given the database a few of them. On the Status menu option, I click on 'composite numbers without known factors' list link. Right now my personal computer is working on an 81 digit number. This C81 might be a semiprime, that is a product of two primes, or possibly a ksmooth number where k is, say, less than 10,000. One of you might know, what are the odds that a C100 is semiprime? Regards, Mattcanderson 
20210814, 14:00  #2  
Feb 2017
Nowhere
3·5·331 Posts 
Quote:
The standard result for products of two primes is Absent any other information, a 100digit number is about 5.4 times as likely to be a P_{2} as it is to be a prime. A significant lower bound on the smallest factor will change the odds. For example, if a C100 has no prime factors less than 10^{100/3}, it is certain to be a P_{2}. 

20210816, 12:45  #3 
Aug 2020
79*6581e4;3*2539e3
398_{10} Posts 
In that regard, the number of distinct prime factors of n can be estimated by 1.38*log(n)/log(log(n)), I think  though that seems quite large for small numbers. Composites < 100 mostly habe less than 3 distinct factors.
For the estimated number of prime factors (not necessarily distinct) I couldn't find a formula, just that it is the big Omega prime function. Is there a similarly simple estimate? What does "standard result" mean? Last fiddled with by Dr Sardonicus on 20210816 at 15:31 Reason: Attributing quote 
20210816, 13:12  #4  
Apr 2020
2^{2}×3×41 Posts 
Quote:


20210816, 15:36  #5  
Feb 2017
Nowhere
3·5·331 Posts 
Quote:
E. Landau, Sur quelques problèmes relatifs à la distribution des numbres premiers, Bull. Soc. Math. France 28 (1900), 25–38. E. Landau, Über die Verteilung der Zahlen, welche aus Primfaktoren zusammengesetzt sind, Gött. Nachr. Math.Phys. Kl. (1911), 361–381. 

20210823, 05:25  #6 
Aug 2020
79*6581e4;3*2539e3
616_{8} Posts 
It was at wikipedia or at stackexchange, in either instance maybe I got something wrong. Thanks for the links!

20210906, 00:31  #7 
"Matthew Anderson"
Dec 2010
Oregon, USA
1101111010_{2} Posts 
Thanks for the informative replies.
Also, I continue to factor composite numbers for factordb.com This is fun for me and useful for anyone who may want these results. Matt Last fiddled with by MattcAnderson on 20210906 at 00:35 Reason: added enthusiasm 
20210906, 07:54  #8 
"Alexander"
Nov 2008
The Alamo City
758_{10} Posts 
I hope and pray that by "factor[ing] composite numbers for factordb.com," you mean submitting factors for numbers already on the site (a great and undervalued service to the factoring community) and not adding brand new numbers with full or partial factorizations (which is considered spam unless the numbers are part of a useful set, like a useful aliquot sequence or numbers of certain special forms, and bloats the database in any case).

20210907, 14:20  #9 
"Matthew Anderson"
Dec 2010
Oregon, USA
2×5×89 Posts 
Yes Happy5214,
I take numbers from under the status menu in factordb.com. These are numbers that factordb.com wants full prime factorizations for. I find full prime factorization and submit my results in the box on the website. Regards, Matt 
20210907, 16:53  #10 
Feb 2017
Nowhere
3×5×331 Posts 
The list of "Composites without known factors" has a bunch of ludicrously small numbers (the smallest is 22 decimal digits IIRC), and page after page of 65 decimal digit numbers of the form (10^70  A)/B.
As an example, I took 1100000002659432902 (10^706240221)/269383 and fed it to the PariGP online calculator. (I was curious if it would take too long or fail for some other reason.) It did have to increase the stack size, but even so, it got the answer: Code:
? factor((10^706240221)/269383) %1 = [1364961305782585979207, 1; 27196278181038383265924982533012091561386259, 1] 
20210914, 10:31  #11  
"Alexander"
Nov 2008
The Alamo City
2·379 Posts 
Quote:
Last fiddled with by Happy5214 on 20210914 at 10:32 Reason: First part kind of a question 

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