# Thread: Primes above googol

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 $101$ divides $10^n+1$ when $n\equiv 2\pmod{4}$... 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 $101$ divides $10^n+1$ when $n$ is divisible by 4... You just need to find what the other factor is.
$101 \not|10^n+1$ when $4|n$

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

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

4. Hello, meshel88!

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

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

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