# String of 1s and Prime numbers

Can you prove that if $\frac{10^{n}-1}{9}$ is a prime number then 'n' is also a prime number?

Note by Vikram Pandya
4 years, 4 months ago

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Umm.. Yes, I can :)

Let's prove the contrapositive. In other words, we're going to prove that if $$n$$ is not a prime number, then $$\frac{10^9-1}{9}$$ will never be a prime number.

Let $$n$$ be equal to $$ab$$ where $$a$$ and $$b$$ are two positive integers greater than $$1$$.

Now notice that

$\frac{10^{ab}-1}{9}=\frac{10^a-1}{9}\times (1+10^a+10^{2a}+10^{3a}+\cdots 10^{a(b-1)})$

which is definitely not a prime number.

A fun fact: primes in the form of $$\frac{10^n-1}{9}$$ are called repunit primes.

- 4 years, 4 months ago

This is true for any base.A repunit is of the form $\dfrac{b^n-1}{b-1}$ in base $$b$$.The above result holds in any base that a repunit is a prime if $$n$$ is a prime.

- 4 years, 4 months ago

Here's a related problem:

How many numbers of the form $$1010\ldots 0101$$ are prime?

Staff - 4 years, 4 months ago

only $$101$$ is a prime

- 4 years ago

Some initial thoughts:

$$101$$ is prime.

$$10101$$ is a multiple of $$3$$. (This holds for all other cases where the number of $$1$$s are a multiple of $$3$$)

$$1010101$$ is the product of at least two factors, namely $$101$$ and $$10001$$. This holds for all greater cases where the number of $$1$$s are even.

Haven't thought of when the number of $$1$$s are odd though.

- 4 years, 4 months ago

When the number of $$1$$s is odd, the number is divisible by the repunit you get by taking out all of the zeroes; e.g. $$10101$$ is divisible by $$111$$, $$101010101$$ is divisible by $$11111$$, and so on. This is because $$1+10^2 + 10^4 + \cdots + 10^{2n} = (1+10+10^2+\cdots+10^n)(1-10+10^2-\cdots+10^n).$$ Note that the last sign in that alternating factor is a $$+$$ only if $$n$$ is odd.

- 4 years, 4 months ago

- 4 years, 4 months ago

Yeah the one way of doing this. I was not sure how to put proof related questions on this forum. But this is good level 4 question otherwise :)

- 4 years, 4 months ago

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