Problems

Look at this formula found by Euler: $n^2+n+41$. It has a remarkable property: for every integer number from $1$ to $21$ it always produces prime numbers. For example, for $n=3$ it is $53$, a prime. For $n=20$ it is $461$, also a prime, and for $n=21$ it is $503$, prime as well. Could it be that this formula produces a prime number for any natural $n$?

Prove that there is a power of 3 that ends in 001.

Prove that if $(m,10)=1$, then there is a repeated unit $E_n$ that is divisible by $m$. Will there be infinitely many repeated units?

Prove that for any odd natural number, $a$, there exists a natural number, $b$, such that $2^b-1$ is divisible by $a$.