Post by quasiPost by barkerPost by Pubkeybreaker10^281-1 is the smallest integer of the form 10^n-1 that is
not fully factored.
No.
It is the smallest one whose prime factors have not been
published. There is a difference...
Reading between the lines, would you like a decomposition of
14975722877914491717620426059118417580707929278605
30935561591990419823492064154065403321443661895485
51670508994053634039049939359298138323011269238802
59832175429893886346932128104884153170949226474007
3694039201
(yes, it has exactly two prime factors).
The problem with that is how do we know that it wasn't
obtained simply by multiplying two large primes?
You have contrived to confuse yourself.
Pubkeybreaker <***@aol.com> mentioned the repunit 10^281-1,
attributing a property incorrectly to it. I never mentioned it until
he did. I am not he. Thereafter, I have provided a large factor of
this number, and can, if properly motivated, provide all its prime
factors (i.e., decompose the number:
149757228779144917176204260591184175807079292786053093556159199041982349
206415406540332144366189548551670508994053634039049939359298138323011269
238802598321754298938863469321281048841531709492264740073694039201 )
Google, Pubkeybreaker's admitted tool of the trade, wouldn't help here.
Divide 10^281-1 by this number, and you will find what I say is true.
The other factor is relatively small and easy to decompose.
I could not, therefore, have arrived at this 200+ digit number simply by
coming up with two primes and multiplying them! It is, instead, a factor
of the number 10^281-1 produced as some form of challenge by the other
poster, Pubkeybreaker <***@aol.com>, who appears, in common
with many AO Lusers, to be a troll. Irrelevant references to the axion
of choice are a way of knowing them by their works.
Post by quasiPost by barkerFor reference, here are all six prime or nearly-prime
113136621886721
61743174809323855441087480850698944226478311618140
3206744221352101186333
69854142224881900093955605694268161454638334297085
131413095647940222173363516839384093
15906159258732342511777147133067716573217266315793
54285924941533406890138660648313890248669058360011
65382255194784215933028413820020114354227
17995691257251674192595206160406625002604393173457
87977520923564818642498503782465461988328080852935
96132153154665017233714517729376660209728956672615
19667
98209677165685617966946301691673463791787856420010
91121947698883162911589290005256072552851310637731
03221026560885703118320809283587891336782742672399
42357743229610071851165137671683769570402913782212
648893179591
Look carefully...
I don't see the pattern (but maybe I didn't look as
closely as I should have).
Let me start you on your voyage of discovery. Please do not assume
this is true except that it is a good point to begin your voyage.
A repunit is a string of "1"s. In decimal notation, the digit-
complement of "1" is "9". "0" has its obvious role within repunit
factor strings.
Look for "9"s in the numbers above, and see how frequently they
are adjacent to "0"s. Look for "1"s, and see how frequently they
are adjacent to "1"s. Then even see how frequently "9"s are
adjacent to "0"s. Assume pseudo-normality, compute what expected
frequencies are, and be startled at how much higher these are.
"90009" - there for sure.
Small sample, I hear? Skewed because I chosed the 227th repunit
(10^227-1)/9? Try it with as many others as you can, at least
partially decomposing them and examining the results of part-
decomposition (warning! consider only prime or near-prime factors,
as defined earlier).
You will soon be a convert.
Just do a frequency expectation chart. It will jump out at you.
Decide whether the patterns found in the prime or nearly-prime
factors of (10^n-1)/9 are stronger (= more frequent) if n is prime.
Should you need assistance with factorizations, there are AO-Lusers
with large computing power and limited brain power (Pubkeybreaker
<***@aol.com> serves well) who can help you, but whose
mathematical intuition is either insignificant or absent.
Working out WHY these factors are far from pseudo-normal, and why
these sequences are so common, now that is another matter altogether.
Post by quasiBut suppose the factors of a repunit have digit patterns
which appear with greater frequency than what would occur
in a random digit string. Now what? Does it cut down the
search space when looking for factors? If so, by how much?
Step by step. Learn first to walk. Consider what would the position
be in a lower notational base. 2 and 3 are no use because the concept
of adjacency breaks down and we are left with having to look for more
complex patterns. In fact, restating all numbers in a much larger base
- for example, base 31 - will illustrate the principle clearer.
There are some deep truths in number theory buried within here.
As appeared Galois, Leibnitz etc., no doubt. To the
Confederacy, that is.
Thank you,
"barker" (associate of the late falsified Dr Pertti Lounesto)