# “Although it is necessary for $n$ to be prime in order for $R_n$ to be prime” as logic statement

This is a statement about Repunits from this paper.

How can I write this as an if/then statement?
Knowledge about Repunits isn't required. The question is basically: what does "although it is necessary for $$a$$ to be prime in order for $$b$$ to be prime" mean?

I think it means: if $$b$$ is prime then $$a$$ is prime.

• "$P$ is necessary for $Q$" (or equivalently, "$Q$ is sufficient for $P$") is, more formally, $Q\to P$. However, linguitically, I find the twofold use of "for" in your quote confusing – Hagen von Eitzen Oct 14 '18 at 20:08

The logically equivalent statement to yours might be:

If $$R_n$$ is prime then $$n$$ is prime.

If you are interested in proving such statement, consider the following "proving the contrapositive":

$$R_n$$ can be written as $$\frac{10^n-1}{9}$$ given that the fraction $$\frac{10^n-1}{9}$$ is always an integer. Now, if $$n$$ is not prime, then $$n = ab$$ for some prime $$a$$ and integer $$b$$. Hence, $$R_n = \frac{10^{ab}-1}{9}=\frac{(10^a)^b-1}{9}$$. But $$(10^a)^b-1 = (10^a - 1) \times \sum_{i=0}^{b-1}(10^a)^{i}$$ and $$R_n = \sum_{i=0}^{b-1}(10^a)^{i} \times \frac{10^a - 1}{9}$$ This implies that $$(\sum_{i=0}^{b-1}(10^a)^{i})$$ is a divisor of the Repunit $$R_n$$. Thereby, if $$n$$ is not prime, then so is $$R_n$$. Contrapositively, $$n$$ must be prime if $$R_n$$ is prime.

although it is necessary for "$$a$$ to be prime" in order for "$$b$$ to be prime".
That is, if $$b$$ is prime, then we know that $$a$$ must be prime.
"If $$b$$ is prime, then $$a$$ is prime" which corresponds to "If $$R_n$$ is prime, then $$n$$ is prime"
Notice that in this case we say that "$$a$$ is prime" is a necessary condition to "$$b$$ is prime". We say that "$$b$$ is prime" is a sufficient condition for "$$a$$ is prime".