Repeating Decimals in Fractions of Prime Numbers

by Bill Price

Fractions of the form **n/p**, where **p** is a prime number
and **n** is a positive integer less than **p**, and **p**
is not a factor of 10 (i.e., the primes 2 and 5), result in repeating decimals with very
definite patterns.

For any prime number **p** above, the set of all repeating decimals
derived from the fractions **1/p, 2/p, 3/p, ... (p-1)/p** is made up of
equal-length "chains" of digits that repeat themselves at fixed intervals. There
can be one or more unique "chains" of digits in this set, depending on the value
of **p**. The length of any one of these unique chains of digits appears to
have a definite relationship to the number of such unique chains for any prime **p**.
This relationship can be described as follows:

If **p** is the prime, and **x** is the length of the chain,
and **y** is the number of unique chains found for all **1/p, 2/p, 3/p,
... (p-1)/p**, then **xy = p-1**.

Note that I define a "unique" chain as a unique sequence of digits that can
be duplicated in another value of **n/p**, beginning with a different digit
of the unique chain, and wrapping around for the same number of digits. For example, **142857**,
**428571**, and **571428** all represent the same unique chain
of six digits for the prime **7**.

Moreover, if **x** is an even number, then the digits of the first half of
the chain are 9's complements of the corresponding digits of the second half of the chain,
that is, they add up to 9.

If **x** is an odd number, then **y** is always even, and the
chains can be grouped into pairs such that all the digits of one chain are 9's complements
of the corresponding digits in the other chain.

Regardless of the length of any chain, whether an odd or even number of digits, the integer that it represents is evenly divisible by 9. This can be seen by successively adding up all digits and the digits of the resulting sums, until a one-digit sum of 9 is obtained.

As the value of **p** increases, the values of **x** and **y**
do not reveal any discernable pattern, other than the relationship given above.

These properties apply to all prime numbers, excluding 2 and 5, and only because they just happen to be factors of 10, the base of our numbering system. Since 10 is evenly divisible by both 2 and 5, neither 1/2 nor 1/5 exhibit the above properties. If, however, our numbering system were based on a prime number (base 17 for instance - see next topic below), then these properties would hold true for all primes (except, of course, for the base prime itself).

These properties of prime decimal chains may be illustrated by the first ten primes (excluding 2 and 5):

Prime: 3

2 chains of 1:

3

6

x = 1, y = 2

The digits complement each other.

3 + 6 = 9.

Prime: 7

1 chain of 6:

142857

x = 6, y = 1

The first three digits 142 complement the last three 857.

142857 / 9 = 15873.

Prime: 11

5 chains of 2:

09

18

27

36

45

x = 2, y = 5

The sum of each pair of digits equals 9.

Prime: 13

2 chains of 6:

076923

153846

x = 6, y = 2

The first three digits of each chain complement the last three.

76923 / 9 = 8547

153846 / 9 = 17094

Prime: 17

1 chain of 16:

0588235294117647

x = 16, y = 1

The first eight digits 05882352 complement the following 94117647.

Sum of digits = 72; 7 + 2 = 9

Prime 19:

1 chain of 18:

052631578947368421

x = 18, y = 1

The first nine digits 052631578 complement the following 947368421.

Sum of digits = 81; 8 + 1 = 9

Prime: 23

1 chain of 22:

0434782608695652173913

x = 22; y = 1

The first eleven digits 04347826086 complement the following 95652173913.

Sum of digits = 99; 9 + 9 = 18; 1 + 8 = 9

Prime: 29

1 chain of 28:

0344827586206896551724137931

x = 28; y = 1

The first fourteen digits 03448275862068 complement the following 96551724137931.

Sum of digits = 126; 1 + 2 + 6 = 9

Prime: 31

2 chains of 15:

032258064516129

967741935483870

x = 15; y = 2

The two chains complement each other.

Sum of digits of first chain = 54; 5 + 4 = 9

Sum of digits of second chain = 81; 8 + 1 = 9

Prime: 37

12 chains of 3:

027

972

054

945

081

918

135

864

162

837

243

756

x = 3; y = 12

Again, all of the chains pair up with other chains to form complements. And again, all of the chains represent number divisible by 9.

Looking ahead to the primes up to 499, we find the same patterns:

Prime | x | y |

41 | 5 | 8 |

43 | 21 | 2 |

47 | 46 | 1 |

53 | 13 | 4 |

59 | 58 | 1 |

61 | 60 | 1 |

67 | 33 | 2 |

71 | 35 | 2 |

73 | 8 | 9 |

79 | 13 | 6 |

83 | 41 | 2 |

89 | 44 | 2 |

97 | 96 | 1 |

101 | 4 | 25 |

103 | 34 | 3 |

107 | 53 | 2 |

109 | 108 | 1 |

113 | 112 | 1 |

127 | 42 | 3 |

131 | 130 | 1 |

137 | 8 | 17 |

139 | 46 | 3 |

149 | 148 | 1 |

151 | 75 | 2 |

157 | 78 | 2 |

163 | 81 | 2 |

167 | 166 | 1 |

173 | 43 | 4 |

179 | 178 | 1 |

181 | 180 | 1 |

191 | 95 | 2 |

193 | 192 | 1 |

197 | 98 | 2 |

199 | 99 | 2 |

211 | 30 | 7 |

223 | 222 | 1 |

227 | 113 | 2 |

229 | 228 | 1 |

233 | 232 | 1 |

239 | 7 | 34 |

241 | 30 | 8 |

251 | 50 | 5 |

257 | 256 | 1 |

263 | 262 | 1 |

269 | 268 | 1 |

271 | 5 | 54 |

277 | 69 | 4 |

281 | 28 | 10 |

283 | 141 | 2 |

293 | 146 | 2 |

307 | 153 | 2 |

311 | 155 | 2 |

313 | 312 | 1 |

317 | 79 | 4 |

331 | 110 | 3 |

337 | 336 | 1 |

347 | 173 | 2 |

349 | 116 | 3 |

353 | 32 | 11 |

359 | 179 | 2 |

367 | 366 | 1 |

373 | 186 | 2 |

379 | 378 | 1 |

383 | 382 | 1 |

389 | 388 | 1 |

397 | 99 | 4 |

401 | 200 | 2 |

409 | 204 | 2 |

419 | 418 | 1 |

421 | 140 | 3 |

431 | 215 | 2 |

433 | 432 | 1 |

439 | 219 | 2 |

443 | 221 | 2 |

449 | 32 | 14 |

457 | 152 | 3 |

461 | 460 | 1 |

463 | 231 | 2 |

467 | 233 | 2 |

479 | 239 | 2 |

487 | 486 | 1 |

491 | 490 | 1 |

499 | 498 | 1 |

We see that for the vast majority of these primes, the value of x is greater than y; in fact, most y values are very low numbers, from 1 to 3. In very few cases does the value y greatly exceed the value x (as in 101, 137, 239, and 271).

These same properties hold true for primes in any numbering system, that is to say,
they are not just phenomena of our base 10 numbering system, but are phenomena unique to
primes themselves, independently of how they are represented in symbols. Let us take, for
instance, a base of **17**, and for the digits that represent the numbers **10**
through **16** we use the letters **A** through **G**.

In base 17 the relationship **xy = p-1** still holds true. However, since
we are now working with base 17 instead of base 10, the digits are **16's**
complements, represented by the symbol **G**. Note also that 1 / 2 and 1 / 5
now step out from under the shadow of base 10 and reveal their inherent
"primeness".

Prime: 2

1 chain of 1:

8

(8 is half of G)

x = 1, y = 1

Prime: 3

1 chain of 2

5B

x = 2, y = 1

The digits complement each other.

5 + B = G

Prime: 5

1 chain of 4:

36DA

x = 4, y = 1

The first two digits 36 complement the following DA.

Sum of digits = 3 + 6 + D + A = 1F; 1 + F = G

Prime: 7

1 chain of 6:

274E9C

x = 6, y = 1

The first three digits 274 complement the following E9C.

Sum of digits = 2 + 7 + 4 + E + 9 + C = 2E; 2 + E = G

Prime: 11

1 chain of 10:

194ADF7C63

x = 10, y = 1

The first three digits 194AD complement the following F7C63.

Sum of digits = 1 + 9 + 4 + A + D + F + 7 + C + 6 + 3 = 4C; 4 + C = G

Prime: 13

2 chains of 6:

153FBD

2A7E69

x = 6, y = 2

The first three digits of each chain complement the last three.

Sum of digits in both cases are multiples of G.

Prime: 19

2 chains of 9:

0F39E5648

G1D72BAC8

x = 9, y = 2

The chains complement each other.

Sum of digits in both cases are multiples of G.

Prime: 23

1 chain of 22:

0C9A5F8ED52G476B1823BE

x = 22, y = 1

The first three digits 0C9A5F8ED52 complement the following G476B1823BE.

Sum of digits is multiple of G.

Prime: 29

7 chains of 4:

09G7

12FE

1CF4

25EB

38D8

4BC5

67A9

x = 4, y = 7

The first two digits of each chain complement the last two.

Sum of digits in all cases are multiples of G.

Prime: 31

1 chain of 30:

09583E469EDC11AG7B8D2CA7234FF6

x = 30, y = 1

The first fifteen digits 09583E469EDC11A complement the following G7B8D2CA7234FF6.

Sum of digits is multiple of G.

Prime: 37

1 chain of 36:

07DD58C6F2CEBG1675G933B84A1E4250FA9B

x = 36, y = 1

The first eighteen digits 07DD58C6F2CEBG1675 complement the following G933B84A1E4250FA9B.

Sum of digits is multiple of G.

Looking ahead to the primes up to 499, we again find the same patterns, but with **x**
and **y** values differing from base 10:

Prime | x | y |

41 | 40 | 1 |

43 | 21 | 2 |

47 | 23 | 2 |

53 | 26 | 2 |

59 | 29 | 2 |

61 | 60 | 1 |

67 | 33 | 2 |

71 | 10 | 7 |

73 | 24 | 3 |

79 | 26 | 3 |

83 | 41 | 2 |

89 | 44 | 2 |

97 | 96 | 1 |

101 | 10 | 10 |

103 | 51 | 2 |

107 | 106 | 1 |

109 | 36 | 3 |

113 | 112 | 1 |

127 | 63 | 2 |

131 | 130 | 1 |

137 | 68 | 2 |

139 | 138 | 1 |

149 | 37 | 4 |

151 | 75 | 2 |

157 | 39 | 4 |

163 | 54 | 3 |

167 | 166 | 1 |

173 | 172 | 1 |

179 | 89 | 2 |

181 | 36 | 5 |

191 | 95 | 2 |

193 | 192 | 1 |

197 | 196 | 1 |

199 | 66 | 3 |

211 | 210 | 1 |

223 | 37 | 6 |

227 | 226 | 1 |

229 | 19 | 12 |

233 | 232 | 1 |

239 | 119 | 2 |

241 | 80 | 3 |

251 | 125 | 2 |

257 | 32 | 8 |

263 | 131 | 2 |

269 | 268 | 1 |

271 | 135 | 2 |

277 | 276 | 1 |

281 | 140 | 2 |

283 | 282 | 1 |

293 | 73 | 4 |

307 | 102 | 3 |

311 | 310 | 1 |

313 | 312 | 1 |

317 | 316 | 1 |

331 | 165 | 2 |

337 | 112 | 3 |

347 | 346 | 1 |

349 | 58 | 6 |

353 | 88 | 4 |

359 | 179 | 2 |

367 | 366 | 1 |

373 | 62 | 6 |

379 | 378 | 1 |

383 | 191 | 2 |

389 | 97 | 4 |

397 | 132 | 3 |

401 | 400 | 1 |

409 | 51 | 8 |

419 | 418 | 1 |

421 | 210 | 2 |

431 | 430 | 1 |

433 | 27 | 16 |

439 | 438 | 1 |

443 | 221 | 2 |

449 | 448 | 1 |

457 | 38 | 12 |

461 | 230 | 2 |

463 | 231 | 2 |

467 | 233 | 2 |

479 | 478 | 1 |

487 | 486 | 1 |

491 | 49 | 10 |

499 | 498 | 1 |

**Chart Survey for Number Bases 11-27**

These following chart surveys the relationships between x and y for number bases from 11 through 28, for the primes 7 through 199. The empty cells appear in those instances where the prime in question is a factor of the number base chosen.

Prime | base 11 x/y | base 12 x/y | base 13 x/y | base 14 x/y | base 15 x/y | base 16 x/y | base 17 x/y | base 18 x/y | base 19 x/y | base 20 x/y | base 21 x/y | base 22 x/y | base 23 x/y | base 24 x/y | base 25 x/y | base 26 x/y | base 27 x/y | base 28 x/y |

7 | 3/2 | 6/1 | 2/3 | - | 1/6 | 3/2 | 6/1 | 3/2 | 6/1 | 2/3 | - | 1/6 | 3/2 | 6/1 | 3/2 | 6/1 | 2/3 | - |

11 | - | 1/10 | 10/1 | 5/2 | 5/2 | 5/2 | 10/1 | 10/1 | 10/1 | 5/2 | 2/5 | - | 1/10 | 10/1 | 5/2 | 5/2 | 5/2 | 10/1 |

13 | 12/1 | 2/6 | - | 1/12 | 12/1 | 3/4 | 6/2 | 4/3 | 12/1 | 12/1 | 4/3 | 3/4 | 6/2 | 12/1 | 2/6 | - | 1/12 | 12/1 |

17 | 16/1 | 16/1 | 4/4 | 16/1 | 8/2 | 2/8 | - | 1/16 | 8/2 | 16/1 | 4/4 | 16/1 | 16/1 | 16/1 | 8/2 | 8/2 | 16/1 | 16/1 |

19 | 3/6 | 6/3 | 18/1 | 18/1 | 18/1 | 9/2 | 9/2 | 2/9 | - | 1/18 | 18/1 | 18/1 | 9/2 | 9/2 | 9/2 | 3/6 | 6/3 | 9/2 |

23 | 22/1 | 11/2 | 11/2 | 22/1 | 22/1 | 11/2 | 22/1 | 11/2 | 22/1 | 22/1 | 22/1 | 2/11 | - | 1/22 | 11/2 | 11/2 | 11/2 | 22/1 |

29 | 28/1 | 4/7 | 14/2 | 28/1 | 28/1 | 7/4 | 4/7 | 28/1 | 28/1 | 7/4 | 28/1 | 14/2 | 7/4 | 7/4 | 7/4 | 28/1 | 28/1 | 2/14 |

31 | 30/1 | 30/1 | 30/1 | 15/2 | 10/3 | 5/6 | 30/1 | 15/2 | 15/2 | 15/2 | 30/1 | 30/1 | 10/3 | 30/1 | 3/10 | 6/5 | 10/3 | 15/2 |

37 | 6/6 | 9/4 | 36/1 | 12/3 | 36/1 | 9/4 | 36/1 | 36/1 | 36/1 | 36/1 | 18/2 | 36/1 | 12/3 | 36/1 | 18/2 | 3/12 | 6/6 | 18/2 |

41 | 40/1 | 40/1 | 40/1 | 8/5 | 40/1 | 5/8 | 40/1 | 5/8 | 40/1 | 20/2 | 20/2 | 40/1 | 10/4 | 40/1 | 10/4 | 40/1 | 8/5 | 40/1 |

43 | 7/6 | 42/1 | 21/2 | 21/2 | 21/2 | 7/6 | 21/2 | 42/1 | 42/1 | 42/1 | 7/6 | 14/3 | 21/2 | 21/2 | 21/2 | 42/1 | 14/3 | 42/1 |

47 | 46/1 | 23/2 | 46/1 | 23/2 | 46/1 | 23/2 | 23/2 | 23/2 | 46/1 | 46/1 | 23/2 | 46/1 | 46/1 | 23/2 | 23/2 | 46/1 | 23/2 | 23/2 |

53 | 26/2 | 52/1 | 13/4 | 52/1 | 13/4 | 13/4 | 26/2 | 52/1 | 52/1 | 52/1 | 52/1 | 52/1 | 4/13 | 13/4 | 26/2 | 52/1 | 52/1 | 13/4 |

59 | 58/1 | 29/2 | 58/1 | 58/1 | 29/2 | 29/2 | 29/2 | 58/1 | 29/2 | 29/2 | 29/2 | 29/2 | 58/1 | 58/1 | 29/2 | 29/2 | 29/2 | 29/2 |

61 | 4/15 | 15/4 | 3/20 | 6/10 | 15/4 | 15/4 | 60/1 | 60/1 | 30/2 | 5/12 | 12/5 | 15/4 | 20/3 | 20/3 | 15/4 | 60/1 | 10/6 | 20/3 |

67 | 66/1 | 66/1 | 66/1 | 11/6 | 11/6 | 33/2 | 33/2 | 66/1 | 33/2 | 66/1 | 33/2 | 11/6 | 33/2 | 11/6 | 11/6 | 33/2 | 22/3 | 66/1 |

71 | 70/1 | 35/2 | 70/1 | 10/7 | 35/2 | 35/2 | 10/7 | 35/2 | 35/2 | 7/10 | 70/1 | 70/1 | 14/5 | 35/2 | 5/14 | 14/5 | 35/2 | 70/1 |

73 | 72/1 | 36/2 | 72/1 | 72/1 | 72/1 | 9/8 | 24/3 | 18/4 | 36/2 | 72/1 | 24/3 | 8/9 | 36/2 | 12/6 | 36/2 | 72/1 | 4/18 | 72/1 |

79 | 39/2 | 26/3 | 39/2 | 26/3 | 26/3 | 39/2 | 26/3 | 13/6 | 39/2 | 39/2 | 13/6 | 13/6 | 3/26 | 6/13 | 39/2 | 39/2 | 26/3 | 78/1 |

83 | 41/2 | 41/2 | 82/1 | 82/1 | 82/1 | 41/2 | 41/2 | 82/1 | 82/1 | 82/1 | 41/2 | 82/1 | 41/2 | 82/1 | 41/2 | 41/2 | 41/2 | 41/2 |

89 | 22/4 | 8/11 | 88/1 | 88/1 | 88/1 | 11/8 | 44/2 | 44/2 | 88/1 | 44/2 | 44/2 | 22/4 | 88/1 | 88/1 | 22/4 | 88/1 | 88/1 | 88/1 |

97 | 48/2 | 16/6 | 96/1 | 96/1 | 96/1 | 12/8 | 96/1 | 16/6 | 32/3 | 32/3 | 96/1 | 4/24 | 96/1 | 24/4 | 48/2 | 96/1 | 16/6 | 32/3 |

101 | 100/1 | 100/1 | 50/2 | 10/10 | 100/1 | 25/4 | 10/10 | 100/1 | 25/4 | 50/2 | 50/2 | 50/2 | 50/2 | 25/4 | 25/4 | 100/1 | 100/1 | 100/1 |

103 | 102/1 | 102/1 | 17/6 | 17/6 | 51/2 | 51/2 | 51/2 | 51/2 | 51/2 | 102/1 | 102/1 | 34/3 | 17/6 | 34/3 | 51/2 | 51/ | 34/3 | 51/2 |

107 | 53/2 | 53/2 | 53/2 | 53/2 | 106/1 | 53/2 | 106/1 | 106/1 | 53/2 | 106/1 | 106/1 | 106/1 | 53/2 | 106/1 | 53/2 | 106/1 | 53/2 | 106/1 |

109 | 108/1 | 54/2 | 108/1 | 108/1 | 27/4 | 9/12 | 36/3 | 108/1 | 36/3 | 54/2 | 27/4 | 27/4 | 36/3 | 108/1 | 27/4 | 27/4 | 9/12 | 54/2 |

113 | 56/2 | 112/1 | 56/2 | 28/4 | 4/28 | 7/16 | 112/1 | 8/14 | 112/1 | 112/1 | 112/1 | 56/2 | 112/1 | 112/1 | 56/2 | 56/2 | 112/1 | 7/16 |

127 | 63/2 | 126/1 | 63/2 | 126/1 | 63/2 | 7/18 | 63/2 | 63/2 | 3/42 | 6/21 | 63/2 | 9/14 | 126/1 | 18/7 | 21/6 | 63/2 | 42/3 | 18/7 |

131 | 65/2 | 65/2 | 65/2 | 130/1 | 65/2 | 65/2 | 130/1 | 26/5 | 26/5 | 65/2 | 65/2 | 130/1 | 130/1 | 26/5 | 65/2 | 130/1 | 65/2 | 65/2 |

137 | 68/2 | 136/1 | 136/1 | 34/4 | 34/4 | 17/8 | 68/2 | 34/4 | 68/2 | 136/1 | 136/1 | 34/4 | 136/1 | 136/1 | 68/2 | 136/1 | 136/1 | 68/2 |

139 | 69/2 | 138/1 | 69/2 | 46/3 | 138/1 | 69/2 | 138/1 | 138/1 | 138/1 | 69/2 | 138/1 | 138/1 | 46/3 | 69/2 | 69/2 | 138/1 | 46/3 | 69/2 |

149 | 148/1 | 148/1 | 148/1 | 148/1 | 148/1 | 37/4 | 37/4 | 148/1 | 37/4 | 74/2 | 148/1 | 74/2 | 148/1 | 74/2 | 37/4 | 74/2 | 148/1 | 37/4 |

151 | 75/2 | 150/1 | 150/1 | 150/1 | 150/1 | 15/10 | 75/2 | 75/2 | 5/30 | 25/6 | 75/2 | 75/2 | 30/5 | 50/3 | 75/2 | 50/3 | 50/3 | 50/3 |

157 | 39/4 | 3/52 | 6/26 | 13/12 | 156/1 | 13/12 | 39/4 | 156/1 | 39/4 | 156/1 | 156/1 | 12/13 | 52/3 | 156/1 | 78/2 | 156/1 | 26/6 | 4/39 |

163 | 162/1 | 162/1 | 54/3 | 81/2 | 81/2 | 81/2 | 54/3 | 162/1 | 162/1 | 162/1 | 27/6 | 27/6 | 18/9 | 81/2 | 27/6 | 81/2 | 54/3 | 54/3 |

167 | 83/2 | 83/2 | 166/1 | 83/2 | 166/1 | 83/2 | 166/1 | 83/2 | 83/2 | 166/1 | 83/2 | 83/2 | 166/1 | 83/2 | 83/2 | 166/1 | 83/2 | 83/2 |

173 | 172/1 | 172/1 | 86/2 | 43/4 | 86/2 | 43/4 | 172/1 | 172/1 | 172/1 | 172/1 | 86/2 | 43/4 | 43/4 | 86/2 | 86/2 | 172/1 | 172/1 | 172/1 |

179 | 178/1 | 89/2 | 89/2 | 89/2 | 89/2 | 89/2 | 89/2 | 178/1 | 89/2 | 89/2 | 178/1 | 89/2 | 178/1 | 178/1 | 89/2 | 178/1 | 89/2 | 178/1 |

181 | 90/2 | 90/2 | 45/4 | 45/4 | 45/4 | 45/4 | 36/5 | 180/1 | 4/45 | 90/2 | 180/1 | 20/9 | 180/1 | 180/1 | 15/12 | 12/15 | 15/12 | 180/1 |

191 | 38/5 | 95/2 | 95/2 | 38/5 | 95/2 | 95/2 | 95/2 | 95/2 | 190/1 | 95/2 | 190/1 | 190/1 | 95/2 | 95/2 | 19/10 | 95/2 | 95/2 | 190/1 |

193 | 64/3 | 24/8 | 64/3 | 32/6 | 192/1 | 24/8 | 192/1 | 96/2 | 192/1 | 64/3 | 48/4 | 192/1 | 32/6 | 32/6 | 96/2 | 192/1 | 16/12 | 48/4 |

197 | 196/1 | 196/1 | 196/1 | 4/49 | 98/2 | 49/4 | 196/1 | 196/1 | 14/14 | 28/7 | 196/1 | 98/2 | 49/4 | 49/4 | 98/2 | 98/2 | 196/1 | 49/4 |

199 | 22/9 | 66/3 | 99/2 | 99/2 | 198/1 | 99/2 | 66/3 | 11/18 | 18/11 | 99/2 | 18/11 | 198/1 | 99/2 | 18/11 | 33/6 | 99/2 | 66/3 | 33/6 |

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