333,667 is Unique Period Prime with Period 9

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Theorem

$333 \, 667$ is a unique period prime whose reciprocal has a period of $9$:

$\dfrac 1 {333 \, 667} = 0 \cdotp \dot 00000 \, 299 \dot 7$


Proof

By long division: 

       0.000002997000002...
      ---------------------
333667)1.000000000000000000
         667334
       --------
         3326660
         3003003
         -------
          3236570
          3003003
          -------
           2335670
           2335669
           -------
                 1000000
                  667334
                 -------
                  ......

It remains to be shown that $333 \, 667$ is the only prime number whose reciprocal has a period of $9$.

We have that:

$333 \, 667 \nmid 10$

From Period of Reciprocal of Prime, the period of such a prime is the order of $10$ modulo $p$.

That is, the smallest integer $d$ such that:

$10^d \equiv 1 \pmod p$


From the above long division we know that the period of $\dfrac 1 {333 \, 667}$ is $9$, so $10^9 \equiv 1 \pmod {333 \, 667}$.

The only other possible primes $p$ whose reciprocals might have a period of $9$ must also satisfy:

$10^9 \equiv 1 \pmod p$

that is:

$p \divides 10^9 - 1$


Consider:

\(\ds 10^9 - 1\) \(=\) \(\ds 999 \, 999 \, 999\)
\(\ds \) \(=\) \(\ds 3^4 \times 37 \times 333 \, 667\) prime factorization

Therefore the only other possible primes whose reciprocals might have a period of $9$ are $3$ and $37$.


From Reciprocal of $37$:

$\dfrac 1 {37} = 0 \cdotp \dot 02 \dot 7$

and trivially:

$\dfrac 1 3 = 0 \cdotp \dot 3$

which has a period of $1$.

As required, the only prime number whose reciprocal has a period of $9$ is $333 \, 667$.

$\blacksquare$


Sources

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