Question

(2+1)(2^2+1)...(2^64+1)-2^128

Answer 1

=1(2+1)(2^2+1)...(2^641)-2^128

=(2-1)(2+1)(2^2+1)(2^4+1)...(2^64+1)-2^128

=(2^2-1)(2^2+1)(2^4+1)...(2^64+1)-2^128.......

=(2^64-1)(2^64+1)-2^128

=2^128-1-2^128

=-1

Answer 2

\(=1.\left(2+1\right)\left(2^2+1\right)...\left(2^{64}+1\right)-2^{128}\)

\(=\left(2-1\right)\left(2+1\right)\left(2^2+1\right)\left(2^4+1\right)...\left(2^{64}+1\right)-2^{128}\)

\(=\left(2^2-1\right)\left(2^2+1\right)\left(2^4+1\right)...\left(2^{64}+1\right)-2^{128}\)

\(...\)

\(=\left(2^{64}-1\right)\left(2^{64}+1\right)-2^{128}\)

\(=2^{128}-1-2^{128}=-1\)