Let Mersenne number 2^{n} -1

if 2^{n} -1 composite

2^{n} -1 = n^{2}xy + (x+y)n + 1

so 2^{n} /n

= (n^{2}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^{11}-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