Mersenne factorization by (nxy+x+y)

[baih]

Let Mersenne number 2ⁿ -1

if 2ⁿ -1 composite



2ⁿ -1 = n²xy + (x+y)n + 1


so 2ⁿ /n

= (n²xy + (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


2¹¹-1 = 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?

sory for my english

[Viliam Furik]

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]

Originally Posted by Viliam Furik

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.

thanks i mean (2^n)-2

[Batalov]

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

[baih]

Originally Posted by Batalov

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

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]

Originally Posted by baih

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

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]

Originally Posted by baih

Let Mersenne number 2ⁿ -1

if 2ⁿ -1 composite



2ⁿ -1 = n²xy + (x+y)n + 1


so 2ⁿ /n

= (n²xy + (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


2¹¹-1 = 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?

sory for my english

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:

https://www.wolframalpha.com/input/?...er+the+integer


https://www.wolframalpha.com/input/?...er+the+integer

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

[LaurV]

Originally Posted by a1call

I think I have a similar post here somewhere.

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.