Reciprocals of Prime Numbers

Theorem
The decimal representations of the reciprocals of the first few prime numbers are as follows:


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! align="right" style = "padding: 2px 10px" | $n$ ! align="left" style = "padding: 2px 10px" | $1 / n$ ! style = "padding: 2px 10px" | Also see
 * align="right" style = "padding: 2px 10px" | $2$
 * align="left" style = "padding: 2px 10px" | $0 \cdotp 5$
 * style = "padding: 2px 10px" |
 * align="right" style = "padding: 2px 10px" | $3$
 * align="left" style = "padding: 2px 10px" | $0 \cdotp \dot 3$
 * style = "padding: 2px 10px" |
 * align="right" style = "padding: 2px 10px" | $5$
 * align="left" style = "padding: 2px 10px" | $0 \cdotp 2$
 * style = "padding: 2px 10px" |
 * align="right" style = "padding: 2px 10px" | $7$
 * align="left" style = "padding: 2px 10px" | $0 \cdotp \dot 14285 \, \dot 7$
 * style = "padding: 2px 10px" | Period of Reciprocal of 7 is of Maximal Length
 * align="right" style = "padding: 2px 10px" | $11$
 * align="left" style = "padding: 2px 10px" | $0 \cdotp \dot 0 \dot 9$
 * style = "padding: 2px 10px" |
 * align="right" style = "padding: 2px 10px" | $13$
 * align="left" style = "padding: 2px 10px" | $0 \cdotp \dot 07692 \dot 3$
 * style = "padding: 2px 10px" | Recurring Parts of Multiples of One Thirteenth
 * align="right" style = "padding: 2px 10px" | $17$
 * align="left" style = "padding: 2px 10px" | $0 \cdotp \dot 05882 \, 35294 \, 11764 \, \dot 7$
 * style = "padding: 2px 10px" | Period of Reciprocal of 17 is of Maximal Length
 * align="right" style = "padding: 2px 10px" | $19$
 * align="left" style = "padding: 2px 10px" | $0 \cdotp \dot 05263 \, 15789 \, 47368 \, 42 \dot 1$
 * style = "padding: 2px 10px" | Period of Reciprocal of 19 is of Maximal Length
 * align="right" style = "padding: 2px 10px" | $23$
 * align="left" style = "padding: 2px 10px" | $0 \cdotp \dot 04347 \, 82608 \, 69565 \, 21739 \, 1 \dot 3$
 * style = "padding: 2px 10px" | Period of Reciprocal of 23 is of Maximal Length
 * align="right" style = "padding: 2px 10px" | $29$
 * align="left" style = "padding: 2px 10px" | $0 \cdotp \dot 03448 \, 27586 \, 20689 \, 65517 \, 24137 \, 93 \dot 1$
 * style = "padding: 2px 10px" |
 * align="right" style = "padding: 2px 10px" | $31$
 * align="left" style = "padding: 2px 10px" | $0 \cdotp \dot 03225 \, 80645 \, 1612 \dot 9$
 * style = "padding: 2px 10px" | Period of Reciprocal of 31 is of Odd Length
 * align="right" style = "padding: 2px 10px" | $37$
 * align="left" style = "padding: 2px 10px" | $0 \cdotp \dot 02 \dot 7$
 * style = "padding: 2px 10px" | Period of Reciprocal of 37 has Length 3
 * align="right" style = "padding: 2px 10px" | $41$
 * align="left" style = "padding: 2px 10px" | $0 \cdotp \dot 0243 \dot 9$
 * style = "padding: 2px 10px" | Period of Reciprocal of 41 has Length 5
 * align="right" style = "padding: 2px 10px" | $43$
 * align="left" style = "padding: 2px 10px" | $0 \cdotp \dot 02325581395348837209 \dot 3$
 * style = "padding: 2px 10px" |
 * align="right" style = "padding: 2px 10px" | $47$
 * align="left" style = "padding: 2px 10px" | $0 \cdotp \dot 021276595744680851063829787234042553191489361 \dot 7$
 * style = "padding: 2px 10px" |
 * align="right" style = "padding: 2px 10px" | $53$
 * align="left" style = "padding: 2px 10px" | $0 \cdotp \dot 018867924528 \dot 3$
 * style = "padding: 2px 10px" | Period of Reciprocal of 53 is of Quarter Maximal Length
 * align="right" style = "padding: 2px 10px" | $59$
 * align="left" style = "padding: 2px 10px" | $0 \cdotp \dot 016949152542372881355932203389830508474576271186440677966 \dot 1$
 * style = "padding: 2px 10px" |
 * align="right" style = "padding: 2px 10px" | $61$
 * align="left" style = "padding: 2px 10px" | $0 \cdotp \dot 01639344262295081967213114754098360655737704918032786885245 \dot 9$
 * style = "padding: 2px 10px" | Reciprocal of 61
 * align="right" style = "padding: 2px 10px" | $67$
 * align="left" style = "padding: 2px 10px" | $0 \cdotp \dot 01492537313432835820895522388059 \dot 7$
 * style = "padding: 2px 10px" |
 * }
 * align="right" style = "padding: 2px 10px" | $47$
 * align="left" style = "padding: 2px 10px" | $0 \cdotp \dot 021276595744680851063829787234042553191489361 \dot 7$
 * style = "padding: 2px 10px" |
 * align="right" style = "padding: 2px 10px" | $53$
 * align="left" style = "padding: 2px 10px" | $0 \cdotp \dot 018867924528 \dot 3$
 * style = "padding: 2px 10px" | Period of Reciprocal of 53 is of Quarter Maximal Length
 * align="right" style = "padding: 2px 10px" | $59$
 * align="left" style = "padding: 2px 10px" | $0 \cdotp \dot 016949152542372881355932203389830508474576271186440677966 \dot 1$
 * style = "padding: 2px 10px" |
 * align="right" style = "padding: 2px 10px" | $61$
 * align="left" style = "padding: 2px 10px" | $0 \cdotp \dot 01639344262295081967213114754098360655737704918032786885245 \dot 9$
 * style = "padding: 2px 10px" | Reciprocal of 61
 * align="right" style = "padding: 2px 10px" | $67$
 * align="left" style = "padding: 2px 10px" | $0 \cdotp \dot 01492537313432835820895522388059 \dot 7$
 * style = "padding: 2px 10px" |
 * }
 * align="right" style = "padding: 2px 10px" | $67$
 * align="left" style = "padding: 2px 10px" | $0 \cdotp \dot 01492537313432835820895522388059 \dot 7$
 * style = "padding: 2px 10px" |
 * }
 * }