simplify (2+1)(2^2+1)(2^4+1)(2^8+1)...(2^32+1)

Question

Simplify the equation

Answer 1

(2+1)(2^2+1)=15=2^4-1

(2+1)(2^2+1)(2^4+1)=255=2^8-1

(2+1)(2^2+1)(2^4+1)(2^8+1)=65535=2^16-1

(2+1)...(2^32+1)=2^64-1=1.84467...*10^19.

Proof by induction:

Postulate Pn=(2+1)...(2^2^(n-1)+1)=2^2^n-1.

So P[n+1]=Pn(2^2^n+1)=(2^2^n-1)(2^2^n+1)=(2^2^n)^2-1=2^2^(n+1)-1.

P1=2^2^1-1=2^2-1=3=2+1.

P2=2^2^2-1=2^4-1=15=(2+1)(2^2+1).

So the postulate holds for n=1 (and 2) and also for P[n+1], therefore by induction the postulate is proven. That means the simplification is simply to apply the proven postulate: the given expression simplifies to 2^64-1.

Answer 2

P2=2^2^2-1=2^4-1=15=(2+1)(2^2+1)