Track / Stage 6 / 151 of 400 #1151 of 1964

Problem 1151

National olympiad, first round

Number theory Difficulty 6.2 Prove it

7. Show that any number of the form $2^{4 n+2}+1$ can be easily factored by the use of the identity $4 x^{4}+1=\left(2 x^{2}+2 x+1\right)\left(2 x^{2}-2 x+1\right)$ . Factor $2^{18}+1$ using this identity.

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

7. $5 \cdot 13 \cdot 37 \cdot 109$

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