(2 + 1) (the square of 2 + 1) (4 of 2 + 1) (8 of 2 + 1) (16 of 2 + 1) (32 of 2 + 1) (64 of 2 + 1) + 1? fast

(2 + 1) (the square of 2 + 1) (4 of 2 + 1) (8 of 2 + 1) (16 of 2 + 1) (32 of 2 + 1) (64 of 2 + 1) + 1? fast

(2 + 1) (square of 2 + 1) (4 of 2 + 1) (8 of 2 + 1) (16 of 2 + 1) (32 of 2 + 1) (64 of 2 + 1) + 1
=(2-1)(2+1)(2^2+1)(2^4+1)(2^8+1)(2^16+1)(2^32+1)(2^64+1)+1
=(2^2-1)(2^2+1)(2^4+1)(2^8+1)(2^16+1)(2^32+1)(2^64+1)+1
=(2^4-1)(2^4+1)(2^8+1)(2^16+1)(2^32+1)(2^64+1)+1
=(2^8-1)(2^8+1)(2^16+1)(2^32+1)(2^64+1)+1
=(2^16-1)(2^16+1)(2^32+1)(2^64+1)+1
=(2^32-1)(2^32+1)(2^64+1)+1
=(2^64-1)(2^64+1)+1
=2^128-1+1
=2^128
2^1=2
2^2=4
2^3=8
2^4=16
2^5=32
.
The mantissa of 2 ^ n is about 2,4,8,6 cycles
128/4=32
So the number at the end of 2 ^ 128 is: 6