1. ## Primes above googol

Prove that 10^100+1 is not the smallest prime greater than googol.

2. Originally Posted by meshel88
Prove that 10^100+1 is not the smallest prime greater than googol.
You can prove that $\displaystyle 101$ divides $\displaystyle 10^n+1$ when $\displaystyle n\equiv 2\pmod{4}$... You just need to find what the other factor is. But this doesn't answer the question... cf. Soroban's post.

3. Originally Posted by Laurent
You can prove that $\displaystyle 101$ divides $\displaystyle 10^n+1$ when $\displaystyle n$ is divisible by 4... You just need to find what the other factor is.
$\displaystyle 101 \not|10^n+1$ when $\displaystyle 4|n$

$\displaystyle 101 \mid10^n+1$ when $\displaystyle n\equiv 2\mod{4}$

Try it out with small $\displaystyle n$ to see for yourself.

4. Hello, meshel88!

Prove that $\displaystyle 10^{100}+1$ is not the smallest prime greater than googol.

Fact: .$\displaystyle 10^{100}+1$ is a composite.

Proof: .$\displaystyle 10^{100}+1 \;=\;\left(10^{20}\right)^5 + 1^5 \;=\; \left(10^{20}+1\right)\left(10^{80} - 10^{60} + 10^{40} - 10^{20} + 1\right)$