# Reciprocals of Prime Numbers

## Theorem

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

$n$ $1 / n$ Also see $2$ $0 \cdotp 5$ $3$ $0 \cdotp \dot 3$ $5$ $0 \cdotp 2$ $7$ $0 \cdotp \dot 14285 \, \dot 7$ Period of Reciprocal of 7 is of Maximal Length $11$ $0 \cdotp \dot 0 \dot 9$ $13$ $0 \cdotp \dot 07692 \dot 3$ $17$ $0 \cdotp \dot 05882 \, 35294 \, 11764 \, \dot 7$ Period of Reciprocal of 17 is of Maximal Length $19$ $0 \cdotp \dot 05263 \, 15789 \, 47368 \, 42 \dot 1$ Period of Reciprocal of 19 is of Maximal Length $23$ $0 \cdotp \dot 04347 \, 82608 \, 69565 \, 21739 \, 1 \dot 3$ Period of Reciprocal of 23 is of Maximal Length $29$ $0 \cdotp \dot 03448 \, 27586 \, 20689 \, 65517 \, 24137 \, 93 \dot 1$ $31$ $0 \cdotp \dot 03225 \, 80645 \, 1612 \dot 9$ Period of Reciprocal of 31 is of Odd Length $37$ $0 \cdotp \dot 02 \dot 7$ Period of Reciprocal of 37 has Length 3

## Sources

- 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): Tables: $6$ The Decimal Reciprocals of the Primes from $7$ to $97$