Track / Stage 4 / 107 of 340 #367 of 1964

Problem 367

AMC 12 late, AIME early

Number theory Difficulty 4.7 Find the answer

Example 9 Calculate
$(2+1)\left(2^{2}+1\right)\left(2^{4}+1\right) \cdots \cdot\left(2^{2^{n}}+1\right) .$

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Official solution

$\begin{array}{l} =1 \times(2+1)\left(2^{2}+1\right)\left(2^{4}+1\right) \cdots \cdots\left(2^{2^{n}}+1\right) \\ =(2-1)(2+1)\left(2^{2}+1\right)\left(2^{4}+1\right) \cdots \cdot\left(2^{2^{n}}+1\right) \\ =\left(2^{2}-1\right)\left(2^{2}+1\right)\left(2^{4}+1\right) \cdots \cdot\left(2^{2^{n}}+1\right) \\ =\cdots \cdots \\ =\left(2^{2^{n}}-1\right)\left(2^{2^{n}}+1\right) \\ =2^{2^{n+1}}-1 . \end{array}$

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.