Is it always true?

Number Theory Level 3

True or false ?

For any positive integer $n$ that cannot be represented as $2^k$ , $10^n+1$ is a composite number.

Details and Assumptions:

$k$ is a whole number.

False True Indeterminate

1 solution

Marco Brezzi
Aug 4, 2017

Write $n$ as $2^ab$ , where

$a,b\in\mathbb{N} \quad 2\nmid b\quad b>1$

So we can factor $10^n+1$ :

$10^n+1=10^{2^ab}+1=\left(10^{2^a}\right)^b+1=\left(10^{2^a}+1\right)\left[\left(10^{2^a}\right)^{b-1}-\left(10^{2^a}\right)^{b-2}+\cdots+1\right]$

That is composite