- #1

- 717

- 184

- Homework Statement
- For any ## n\geq 1 ##, prove that there exists a prime with at least ## n ## of its digits equal to ## 0 ##.

[Hint: Consider the arithmetic progression ## 10^{n+1}k+1 ## for ## k=1, 2, ##....]

- Relevant Equations
- None.

Proof:

By Dirichlet's theorem, we have that if ## a ## and ## d ## are two positive coprime numbers,

then there are infinitely many primes of the form ## a+nd ## for some ## n\in\mathbb{N} ##.

Let ## n\geq 1 ## be a natural number.

Now we consider the arithmetic progression ## 10^{n+1}k+1 ## for some ## k\in\mathbb{N} ##.

Then ## a=10^{n+1} ## and ## d=1 ##.

This means ## gcd(a, d)=1 ## where ## a ## and ## d ## are coprime numbers.

Thus, ## 10^{n+1}k+1 ## has at least ## n ## consecutive zeros for every ## k ##.

Therefore, there exists a prime with at least ## n ## of its digits equal to ## 0 ## for any ## n\geq 1 ##.

By Dirichlet's theorem, we have that if ## a ## and ## d ## are two positive coprime numbers,

then there are infinitely many primes of the form ## a+nd ## for some ## n\in\mathbb{N} ##.

Let ## n\geq 1 ## be a natural number.

Now we consider the arithmetic progression ## 10^{n+1}k+1 ## for some ## k\in\mathbb{N} ##.

Then ## a=10^{n+1} ## and ## d=1 ##.

This means ## gcd(a, d)=1 ## where ## a ## and ## d ## are coprime numbers.

Thus, ## 10^{n+1}k+1 ## has at least ## n ## consecutive zeros for every ## k ##.

Therefore, there exists a prime with at least ## n ## of its digits equal to ## 0 ## for any ## n\geq 1 ##.