333,667 is Only Prime whose Reciprocal is of Period 9

From ProofWiki
Jump to: navigation, search

Theorem

The only prime number whose reciprocal has a period of $9$ is $333 \, 667$:

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


Proof

By long division:

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



Sources