`[1+(2+1)(2^2+1)(2^4+1)(2^8+1)(2^16+1)(2^32+1)]=?`

Options

A. `2^(64)-1`
B. `2^(64)+1`
C. `2^(64)`
D. `2^(128)`

Answer

To solve the expression \(1 + (2 + 1)(2^2 + 1)(2^4 + 1)(2^8 + 1)(2^{16} + 1)(2^{32} + 1)\), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ 1 + (2 + 1)(2^2 + 1)(2^4 + 1)(2^8 + 1)(2^{16} + 1)(2^{32} + 1) \]
This can be simplified to:
\[ 1 + 3 \cdot 5 \cdot 17 \cdot 257 \cdot 65537 \cdot (2^{32} + 1) \]

### Step 2: Recognize the pattern
Notice that each term in the product can be expressed in the form \(2^n + 1\). We can use the identity:
\[ (2^n - 1)(2^n + 1) = 2^{2n} - 1 \]
This means that:
\[ (2 + 1)(2^2 + 1)(2^4 + 1)(2^8 + 1)(2^{16} + 1)(2^{32} + 1) = \frac{2^{64} - 1}{2 - 1} = 2^{64} - 1 \]

### Step 3: Substitute back into the expression
Now substituting back into our expression, we have:
\[ 1 + (2^{64} - 1) = 2^{64} \]

### Step 4: Final result
Thus, the final result is:
\[ \boxed{2^{64}} \]