mersenneforum.org (https://www.mersenneforum.org/index.php)
-   Miscellaneous Math (https://www.mersenneforum.org/forumdisplay.php?f=56)
-   -   factoring 2ⁿ-2 equivalent to factoring 2ⁿ-1(I think) (https://www.mersenneforum.org/showthread.php?t=25958)

 baih 2020-09-13 18:03

factoring 2ⁿ-2 equivalent to factoring 2ⁿ-1(I think)

2[SUP]n[/SUP]1= a*b

factoring 2[SUP]n[/SUP]-2 equivalent to factoring 2[SUP]n[/SUP]-1

if this condition is met (a-1) divided by (b-1)
then 2[SUP]n[/SUP]-1 =1 mod b-1

example
M11 = 0 mod 23 and 88 =0 mod 22
2[SUP]11[/SUP]-2 = 0 mod 22

M23 = 0 mod 47
2[SUP]23[/SUP]-2 = 0 mod 46

M73 = 0 mod 439
2[SUP]73[/SUP]-2 = 0 mod 438

M83 = 0 mod 167
2[SUP]83[/SUP]-2 = 0 mod 166

M131 = 0 mod 263
2[SUP]131[/SUP]-2 = 0 mod 262

 paulunderwood 2020-09-13 18:25

[QUOTE=baih;556894]
8<-----SNIP------

M29 = 0 mod 233
2[SUP]29[/SUP]-2 = 0 mod 232

8<-----SNIP------
[/QUOTE]

[CODE](2^29-2)%232
174
[/CODE]

:ermm:

 baih 2020-09-13 18:42

JUST arror because 2304167 MOD 232 not equal to n1
SO sory 29 is false

2[SUP]n[/SUP]-1= [B]a*b[/B] [COLOR="Black"][B] this condition is IF (a-1) divided by (b-1)[/B][/COLOR]

 Dr Sardonicus 2020-09-14 13:16

If p is prime then 2[sup]p[/sup] - 2 is divisible by 2p. So if q divides 2[sup]p[/sup] - 1, gcd(q-1, 2[sup]p[/sup] - 2) is divisible by 2p.

However, 2[sup]n[/sup] - 2 is divisible by 2 but not by 4 if n > 1. If p is prime and the smallest prime factor q of 2[sup]p[/sup] - 1 is congruent to 1 (mod 4), q-1 does not divide 2[sup]p[/sup] - 2.

Consulting a table of factorizations of 2[sup]n[/sup] - 1, I find that p = 29 (already noted) is the smallest p with this property. The next is p = 53 (q = 6361).

 baih 2020-09-17 22:30

greatest common factor
its clear that for a composite numbre n = ab

n-1 divide gcf(a-1,b-1)

if gcf are a big numbre we can use it for factoring n

this what make the contruction of RSA numbre (beware) of making gcf small and does not help for factoring

example

n= 14111 = 103* 137
14110 = 0 MOD 34

and of course gcf(102,136) = 34

 Uncwilly 2020-09-17 22:59

So

30526410117453380369827961 is a semi-prime
gcf(a-1,b-1) = 8 a free fact granted by me.

n-1 = 2^3×5×17×5827×1288671127×5978327543
so the gcf could be any combination of 1 or more of those.

 baih 2020-09-17 23:06

no
becaus 8 very smal
i said earlier it needs to be very large to help

 mathwiz 2020-09-18 00:03

[QUOTE=baih;557258]no
becaus 8 very smal
i said earlier it needs to be very large to help[/QUOTE]

[url]http://factordb.com/index.php?query=2%5E1277-2[/url] is fully factored. Are you saying you can therefore factor 2^1277-1?

 baih 2020-09-18 01:51

I DONT try to facto 2^1277-1 BY 2^1277-2
because i know the possibilty to find a large gcf(a-1,b-1) is very very small (excluded)
approximately the gcf < 10 digts it wont do anything if the factor is large

but this does not mean that the method is incorrect

 JeppeSN 2020-09-21 07:11

[QUOTE=mathwiz;557260][url]http://factordb.com/index.php?query=2%5E1277-2[/url] is fully factored. Are you saying you can therefore factor 2^1277-1?[/QUOTE]
I see an easy factorization of 2^n - 1 by induction: Suppose 2^(n-1) - 1 is factored. Then the factorization of 2^n - 2 is trivial. If we could somehow get the factorization from 2^n - 1 from that, ... /JeppeSN

 All times are UTC. The time now is 00:46.