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" | $1 / n$ ! align="left" style = "padding: 2px 10px" | Period ! style = "padding: 2px 10px" | Also see
 * align="right" style = "padding: 2px 10px" | $2$
 * align="left" style = "padding: 2px 10px" | $0 \cdotp 5$
 * align="right" style = "padding: 2px 10px" | $0$
 * style = "padding: 2px 10px" | Reciprocal of $2$
 * align="right" style = "padding: 2px 10px" | $3$
 * align="left" style = "padding: 2px 10px" | $0 \cdotp \dot 3$
 * align="right" style = "padding: 2px 10px" | $1$
 * style = "padding: 2px 10px" | Reciprocal of $3$
 * align="right" style = "padding: 2px 10px" | $5$
 * align="left" style = "padding: 2px 10px" | $0 \cdotp 2$
 * align="right" style = "padding: 2px 10px" | $0$
 * style = "padding: 2px 10px" | Reciprocal of $5$
 * align="right" style = "padding: 2px 10px" | $7$
 * align="left" style = "padding: 2px 10px" | $0 \cdotp \dot 14285 \, \dot 7$
 * align="right" style = "padding: 2px 10px" | $6$
 * style = "padding: 2px 10px" | Reciprocal of $7$
 * align="right" style = "padding: 2px 10px" | $11$
 * align="left" style = "padding: 2px 10px" | $0 \cdotp \dot 0 \dot 9$
 * align="right" style = "padding: 2px 10px" | $2$
 * style = "padding: 2px 10px" | Reciprocal of $11$
 * align="right" style = "padding: 2px 10px" | $13$
 * align="left" style = "padding: 2px 10px" | $0 \cdotp \dot 07692 \, \dot 3$
 * align="right" style = "padding: 2px 10px" | $6$
 * style = "padding: 2px 10px" | Reciprocal of $13$
 * align="right" style = "padding: 2px 10px" | $17$
 * align="left" style = "padding: 2px 10px" | $0 \cdotp \dot 05882 \, 35294 \, 11764 \, \dot 7$
 * align="right" style = "padding: 2px 10px" | $16$
 * style = "padding: 2px 10px" | Reciprocal of $17$
 * align="right" style = "padding: 2px 10px" | $19$
 * align="left" style = "padding: 2px 10px" | $0 \cdotp \dot 05263 \, 15789 \, 47368 \, 42 \dot 1$
 * align="right" style = "padding: 2px 10px" | $18$
 * style = "padding: 2px 10px" | Reciprocal of $19$
 * align="right" style = "padding: 2px 10px" | $23$
 * align="left" style = "padding: 2px 10px" | $0 \cdotp \dot 04347 \, 82608 \, 69565 \, 21739 \, 1 \dot 3$
 * align="right" style = "padding: 2px 10px" | $22$
 * style = "padding: 2px 10px" | Reciprocal of $23$
 * align="right" style = "padding: 2px 10px" | $29$
 * align="left" style = "padding: 2px 10px" | $0 \cdotp \dot 03448 \, 27586 \, 20689 \, 65517 \, 24137 \, 93 \dot 1$
 * align="right" style = "padding: 2px 10px" | $28$
 * style = "padding: 2px 10px" | Reciprocal of $29$
 * align="right" style = "padding: 2px 10px" | $31$
 * align="left" style = "padding: 2px 10px" | $0 \cdotp \dot 03225 \, 80645 \, 1612 \dot 9$
 * align="right" style = "padding: 2px 10px" | $15$
 * style = "padding: 2px 10px" | Reciprocal of $31$
 * align="right" style = "padding: 2px 10px" | $37$
 * align="left" style = "padding: 2px 10px" | $0 \cdotp \dot 02 \dot 7$
 * align="right" style = "padding: 2px 10px" | $3$
 * style = "padding: 2px 10px" | Reciprocal of $37$
 * align="right" style = "padding: 2px 10px" | $41$
 * align="left" style = "padding: 2px 10px" | $0 \cdotp \dot 0243 \dot 9$
 * align="right" style = "padding: 2px 10px" | $5$
 * style = "padding: 2px 10px" | Reciprocal of $41$
 * align="right" style = "padding: 2px 10px" | $43$
 * align="left" style = "padding: 2px 10px" | $0 \cdotp \dot 02325 \, 58139 \, 53488 \, 37209 \, \dot 3$
 * align="right" style = "padding: 2px 10px" | $21$
 * style = "padding: 2px 10px" | Reciprocal of $43$
 * align="right" style = "padding: 2px 10px" | $47$
 * align="left" style = "padding: 2px 10px" | $0 \cdotp \dot 02127 \, 65957 \, 44680 \, 85106 \, 38297 \, 87234 \, 04255 \, 31914 \, 89361 \, \dot 7$
 * align="right" style = "padding: 2px 10px" | $46$
 * style = "padding: 2px 10px" | Reciprocal of $47$
 * align="right" style = "padding: 2px 10px" | $53$
 * align="left" style = "padding: 2px 10px" | $0 \cdotp \dot 01886 \, 79245 \, 28 \dot 3$
 * align="right" style = "padding: 2px 10px" | $13$
 * style = "padding: 2px 10px" | Reciprocal of $53$
 * align="right" style = "padding: 2px 10px" | $59$
 * align="left" style = "padding: 2px 10px" | $0 \cdotp \dot 01694 \, 91525 \, 42372 \, 88135 \, 59322 \, 03389 \, 83050 \, 84745 \, 76271 \, 18644 \, 06779 \, 66 \dot 1$
 * align="right" style = "padding: 2px 10px" | $58$
 * style = "padding: 2px 10px" |Reciprocal of $59$
 * align="right" style = "padding: 2px 10px" | $61$
 * align="left" style = "padding: 2px 10px" | $0 \cdotp \dot 01639 \, 34426 \, 22950 \, 81967 \, 21311 \, 47540 \, 98360 \, 65573 \, 77049 \, 18032 \, 78688 \, 5245 \dot 9$
 * align="right" style = "padding: 2px 10px" | $60$
 * style = "padding: 2px 10px" | Reciprocal of $61$
 * align="right" style = "padding: 2px 10px" | $67$
 * align="left" style = "padding: 2px 10px" | $0 \cdotp \dot 01492 \, 53731 \, 34328 \, 35820 \, 89552 \, 23880 \, 59 \dot 7$
 * align="right" style = "padding: 2px 10px" | $33$
 * style = "padding: 2px 10px" | Reciprocal of $67$
 * align="right" style = "padding: 2px 10px" | $71$
 * align="left" style = "padding: 2px 10px" | $0 \cdotp \dot 01408 \, 45070 \, 42253 \, 52112 \, 67605 \, 63380 \, 2816 \dot 9$
 * align="right" style = "padding: 2px 10px" | $35$
 * style = "padding: 2px 10px" | Reciprocal of $71$
 * align="right" style = "padding: 2px 10px" | $73$
 * align="left" style = "padding: 2px 10px" | $0 \cdotp \dot 01369 \, 86 \dot 3$
 * align="right" style = "padding: 2px 10px" | $8$
 * style = "padding: 2px 10px" | Reciprocal of $73$
 * align="right" style = "padding: 2px 10px" | $79$
 * align="left" style = "padding: 2px 10px" | $0 \cdotp \dot 01265 \, 82278 \, 48 \dot 1$
 * align="right" style = "padding: 2px 10px" | $13$
 * style = "padding: 2px 10px" | Reciprocal of $79$
 * align="right" style = "padding: 2px 10px" | $83$
 * align="left" style = "padding: 2px 10px" | $0 \cdotp \dot 01204 \, 81927 \, 71084 \, 33734 \, 93975 \, 90361 \, 44578 \, 31325 \, \dot 3$
 * align="right" style = "padding: 2px 10px" | $41$
 * style = "padding: 2px 10px" | Reciprocal of $83$
 * align="right" style = "padding: 2px 10px" | $89$
 * align="left" style = "padding: 2px 10px" | $0 \cdotp \dot 01123 \, 59550 \, 56179 \, 77528 \, 08988 \, 76404 \, 49438 \, 20224 \, 719 \dot 1$
 * align="right" style = "padding: 2px 10px" | $44$
 * style = "padding: 2px 10px" | Reciprocal of $89$
 * align="right" style = "padding: 2px 10px" | $97$
 * align="left" style = "padding: 2px 10px" | $0 \cdotp \dot 01030 \, 92783 \, 50515 \, 46391 \, 75257 \, 73195 \, 87628 \, 86597 \, 93814 \, 43298 \, 96907 \, 21649 \, 48453 \, 60824 \, 74226 \, 80412 \, 37113 \, 40206 \, 18556 \, \dot 7$
 * align="right" style = "padding: 2px 10px" | $96$
 * style = "padding: 2px 10px" | Reciprocal of $97$
 * }
 * style = "padding: 2px 10px" | Reciprocal of $71$
 * align="right" style = "padding: 2px 10px" | $73$
 * align="left" style = "padding: 2px 10px" | $0 \cdotp \dot 01369 \, 86 \dot 3$
 * align="right" style = "padding: 2px 10px" | $8$
 * style = "padding: 2px 10px" | Reciprocal of $73$
 * align="right" style = "padding: 2px 10px" | $79$
 * align="left" style = "padding: 2px 10px" | $0 \cdotp \dot 01265 \, 82278 \, 48 \dot 1$
 * align="right" style = "padding: 2px 10px" | $13$
 * style = "padding: 2px 10px" | Reciprocal of $79$
 * align="right" style = "padding: 2px 10px" | $83$
 * align="left" style = "padding: 2px 10px" | $0 \cdotp \dot 01204 \, 81927 \, 71084 \, 33734 \, 93975 \, 90361 \, 44578 \, 31325 \, \dot 3$
 * align="right" style = "padding: 2px 10px" | $41$
 * style = "padding: 2px 10px" | Reciprocal of $83$
 * align="right" style = "padding: 2px 10px" | $89$
 * align="left" style = "padding: 2px 10px" | $0 \cdotp \dot 01123 \, 59550 \, 56179 \, 77528 \, 08988 \, 76404 \, 49438 \, 20224 \, 719 \dot 1$
 * align="right" style = "padding: 2px 10px" | $44$
 * style = "padding: 2px 10px" | Reciprocal of $89$
 * align="right" style = "padding: 2px 10px" | $97$
 * align="left" style = "padding: 2px 10px" | $0 \cdotp \dot 01030 \, 92783 \, 50515 \, 46391 \, 75257 \, 73195 \, 87628 \, 86597 \, 93814 \, 43298 \, 96907 \, 21649 \, 48453 \, 60824 \, 74226 \, 80412 \, 37113 \, 40206 \, 18556 \, \dot 7$
 * align="right" style = "padding: 2px 10px" | $96$
 * style = "padding: 2px 10px" | Reciprocal of $97$
 * }
 * align="right" style = "padding: 2px 10px" | $97$
 * align="left" style = "padding: 2px 10px" | $0 \cdotp \dot 01030 \, 92783 \, 50515 \, 46391 \, 75257 \, 73195 \, 87628 \, 86597 \, 93814 \, 43298 \, 96907 \, 21649 \, 48453 \, 60824 \, 74226 \, 80412 \, 37113 \, 40206 \, 18556 \, \dot 7$
 * align="right" style = "padding: 2px 10px" | $96$
 * style = "padding: 2px 10px" | Reciprocal of $97$
 * }