Let $n$ be a nature number is relatively prime to 10,such the period of the decimal expansion of $1/n$ is $n-1$ or a divisor of $n-1$, and let $c$ be the "cycle length of $n$" (defined below). If $n-1=2^xc$ or $n+1=2^xc$ for some $x\in\Bbb{N}^{>0}$, then $n$ must be prime.

For example, using $n=41$:

The period of $1/41=0.0243902439\dots$ is $5$ and $5$ is a divisor of $41-1$, the "cycle length of $41$" is $10$, $(41-1)/10 = 4 = 2^2$, so $41$ must be prime.

Is there any counterexample to this?

To define the "cycle length of $n$" (using $n=73$ as an example):

```
Step 1 : 73 + 1 = 74. Get the odd part of 74, which is 37
Step 2 : 73 + 37 = 110. Get the odd part of 110, which is 55
Step 3 : 73 + 55 = 128. Get the odd part of 128, which is 1
```

Continuing this operation (with $73 + 1$) repeats the same steps as above. There are $3$ steps in the cycle, so the cycle length of $73$ is $c=3$.