Reciprocals of Prime Numbers

Theorem

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

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

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): Tables: $6$ The Decimal Reciprocals of the Primes from $7$ to $97$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): Tables: $6$ The Decimal Reciprocals of the Primes from $7$ to $97$