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-   -   Mersenne factorization by (nxy+x+y) (https://www.mersenneforum.org/showthread.php?t=25855)

 baih 2020-08-20 20:47

Mersenne factorization by (nxy+x+y)

[CENTER]Let Mersenne number 2[SUP]n[/SUP] -1

if 2[SUP]n[/SUP] -1 composite

2[SUP]n[/SUP] -1 = n[SUP]2[/SUP]xy + (x+y)n + 1

so 2[SUP]n[/SUP] /n

= (n[SUP]2[/SUP]xy + (x+y)n) /n

= nxy+x+y

Finding the x and y

we can factor the number into a product (nx)+1 and (ny)+1

example

2[SUP]11[/SUP]-1 = 2047
(2047-1) /2= 186

186 = nxy+x+y
= 11* 8*2 + 8+2
X= 8
Y=2
and 2047 = (88+1)*(22+1)

Difficulty and complexity

(nxy+x+y) like a Diophantine equation

Are there any solutions?[/CENTER]

sory for my english

 Viliam Furik 2020-08-20 22:20

Obvious mistakes

I would like to point out a few mistakes.

1. (2^n) is never divisible by (n), (2^n-2) is divisible by (n), when (n) is prime (btw, it is because of Little Fermat theorem)
2. (2047-1) /2= 186; you probably meant (2047-1)/11 = 186.

Apart from these typos, I guess I will leave the topic for other guys.

 baih 2020-08-20 22:27

[QUOTE=Viliam Furik;554433]I would like to point out a few mistakes.

1. (2^n) is never divisible by (n), (2^n-2) is divisible by (n), when (n) is prime (btw, it is because of Little Fermat theorem)
2. (2047-1) /2= 186; you probably meant (2047-1)/11 = 186.

Apart from these typos, I guess I will leave the topic for other guys.[/QUOTE]

thanks i mean (2^n)-2

 Batalov 2020-08-21 00:38

Please demonstrate the power of this method on a tiny number 2^1277-1.
it is composite.
Show us.

 baih 2020-08-21 01:09

[QUOTE=Batalov;554448]Please demonstrate the power of this method on a tiny number 2^1277-1.
it is composite.
Show us.[/QUOTE]

non

The difficulty is the same as the difficulty of (Trial division)
But it may help in some cases

If someone found a solution to the equation c=nxy+x+y

 mathwiz 2020-08-21 03:21

[QUOTE=baih;554454]If someone found a solution to the equation c=nxy+x+y[/QUOTE]

There's infinitely many solutions: c=x=y=0, x=y=1 and c=n+2, and so on.

What is the purpose of this is equation and what constraints are you placing on the variables?

 a1call 2020-08-21 03:42

[QUOTE=baih;554421][CENTER]Let Mersenne number 2[SUP]n[/SUP] -1

if 2[SUP]n[/SUP] -1 composite

2[SUP]n[/SUP] -1 = n[SUP]2[/SUP]xy + (x+y)n + 1

so 2[SUP]n[/SUP] /n

= (n[SUP]2[/SUP]xy + (x+y)n) /n

= nxy+x+y

Finding the x and y

we can factor the number into a product (nx)+1 and (ny)+1

example

2[SUP]11[/SUP]-1 = 2047
(2047-1) /2= 186

186 = nxy+x+y
= 11* 8*2 + 8+2
X= 8
Y=2
and 2047 = (88+1)*(22+1)

Difficulty and complexity

(nxy+x+y) like a Diophantine equation

Are there any solutions?[/CENTER]

sory for my english[/QUOTE]
That's a good find. I think I have a similar post here somewhere.

The problem is you need brute-force (trying different integers for a solution) and the combinations are astronomically large.

You might have some fun with Wolfram-Alpha:

[url]https://www.wolframalpha.com/input/?i=186+%3D+11xy%2Bx%2By+solve+over+the+integer[/url]

[url]https://www.wolframalpha.com/input/?i=%28%282%5E1277-1%29-1%29+%2F2%3D+1277xy%2Bx%2By+solve+over+the+integer[/url]

Good luck, try expanding the concept. You might get something interesting or at worst expand your thinking-power in the process.

 LaurV 2020-08-25 07:55

[QUOTE=a1call;554470]I think I have a similar post here somewhere.[/QUOTE]Except your post was left-aligned, therefore easier to read, haha. This is just some rubbish thrown in the middle of the screen, impossible to read.

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