[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