Comparative base maximum periods

(* indicates the expansion of the reciprocal terminates)

b	2-1	3-1	4-1	5-1	6-1	7-1	8-1	9-1	10-1	11-1	12-1	13-1	14-1	15-1	16-1	17-1	18-1	19-1	20-1	21-1	22-1	Max.	#>4	#>5	#>6	#>8	#>9	#>10
2	*	2	*	4	2	3	*	6	4	10	2	12	3	4	*	8	6	18	4	6	10	18	8	8	5	4	4	2
3	1	*	2	4	1	6	2	*	4	5	2	3	6	4	4	16	1	18	4	6	5	18	7	5	2	2	2	2
4	*	1	*	2	1	3	*	3	2	5	1	6	3	2	*	4	3	9	2	3	5	9	4	2	1	1	0	0
5	1	2	1	*	2	6	2	6	1	5	2	4	6	2	4	16	6	9	1	6	5	16	9	7	2	2	1	1
6	*	*	*	1	*	2	*	*	1	10	*	12	2	1	*	16	*	9	1	2	10	16	5	5	5	5	4	2
7	1	1	2	4	1	*	2	3	4	10	2	12	1	4	2	16	3	3	4	1	10	16	4	4	4	4	4	2
8	*	2	*	4	2	1	*	2	4	10	2	4	1	4	*	8	2	6	4	2	10	10	4	4	3	2	2	0
9	1	*	1	2	1	3	1	*	2	5	1	3	3	2	2	8	1	9	2	3	5	9	4	2	2	1	0	0
10	*	1	*	*	1	6	*	1	*	2	1	6	6	1	*	16	1	18	*	6	2	18	6	6	2	2	2	2
11	1	2	2	1	2	3	2	6	1	*	2	12	3	2	4	16	6	3	2	6	1	16	5	5	2	2	2	2
12	*	*	*	4	*	6	*	*	4	1	*	2	6	4	*	16	*	6	4	6	1	16	5	5	1	1	1	1
13	1	1	1	4	1	2	2	3	4	10	1	*	2	4	4	4	3	18	4	2	10	18	3	3	3	3	3	1
14	*	2	*	2	2	*	*	6	2	5	2	1	*	2	*	16	6	18	2	2	5	18	6	4	2	2	2	2
15	1	*	2	*	1	1	2	*	1	5	2	12	1	*	2	8	1	18	2	1	5	18	5	3	3	2	2	2
16	*	1	*	1	1	3	*	3	1	5	1	3	3	1	*	2	3	9	1	3	5	9	3	1	1	1	0	0
17	1	2	1	4	2	6	1	2	4	10	2	6	6	4	1	*	2	9	4	6	10	10	7	7	3	3	2	0
18	*	*	*	4	*	3	*	*	4	10	*	4	3	4	*	1	*	2	4	3	10	10	2	2	2	2	2	0
19	1	1	2	2	1	6	2	1	2	10	2	12	6	2	4	8	1	*	2	6	10	12	7	7	4	3	3	1
20	*	2	*	*	2	2	*	6	*	5	2	12	2	2	*	16	6	1	*	2	5	16	6	4	2	2	2	2
21	1	*	1	1	1	*	2	*	1	2	1	4	1	1	4	4	1	18	1	*	2	18	1	1	1	1	1	1
22	*	1	*	4	1	1	*	3	4	*	1	3	1	4	*	16	3	18	4	1	*	18	2	2	2	2	2	2
23	1	2	2	4	2	3	2	6	4	1	2	6	3	4	2	16	6	9	4	6	1	16	6	6	2	2	1	1
24	*	*	*	2	*	6	*	*	2	10	*	12	6	2	*	16	*	9	2	6	10	16	8	8	5	5	4	2
25	1	1	1	*	1	3	1	3	1	5	1	2	3	1	2	8	3	9	1	3	5	9	4	2	2	1	0	0
26	*	2	*	1	2	6	*	2	1	5	2	*	6	2	*	8	2	3	1	6	5	8	6	4	1	0	0	0
27	1	*	2	4	1	2	2	*	4	5	2	1	2	4	4	16	1	6	4	2	5	16	4	2	1	1	1	1
28	*	1	*	4	1	*	*	1	4	10	1	12	*	4	*	16	1	9	4	1	10	16	5	5	5	5	4	2
29	1	2	1	2	2	1	2	6	2	10	2	3	1	2	4	16	6	18	2	2	10	18	6	6	4	4	4	2
30	*	*	*	*	*	3	*	*	*	10	*	6	3	*	*	4	*	3	*	3	10	10	3	3	2	2	2	0
31	1	1	2	1	1	6	2	3	1	5	2	4	6	1	2	16	3	6	2	6	5	16	7	5	1	1	1	1
32	*	2	*	4	2	3	*	6	4	2	2	12	3	4	*	8	6	18	4	6	2	18	6	6	3	2	2	2
33	1	*	1	4	1	6	1	*	4	*	1	12	6	4	1	2	1	18	4	6	1	18	5	5	2	2	2	2
34	*	1	*	2	1	2	*	3	2	1	1	4	2	2	*	*	3	18	2	2	1	18	1	1	1	1	1	1
35	1	2	2	*	2	*	2	2	1	10	2	3	1	2	4	1	2	9	2	2	10	10	3	3	3	3	2	0
36	*	*	*	1	*	1	*	*	1	5	*	6	1	1	*	8	*	9	1	1	5	9	5	3	2	1	0	0
37	1	1	1	4	1	3	2	1	4	5	1	12	3	4	4	16	1	2	4	3	5	16	4	2	2	2	2	2
38	*	2	*	4	2	6	*	6	4	5	2	2	6	4	*	4	6	*	4	6	5	6	7	5	0	0	0	0
39	1	*	2	2	1	3	2	*	2	10	2	*	3	2	2	16	1	1	2	3	10	16	3	3	3	3	3	1
40	*	1	*	*	1	6	*	3	*	10	1	1	6	1	*	16	3	18	*	6	10	18	7	7	4	4	4	2
41	1	2	1	1	2	2	1	6	1	10	2	12	2	2	2	16	6	18	1	2	10	18	7	7	5	5	5	3
42	*	*	*	4	*	*	*	*	4	5	*	3	*	4	*	8	*	9	4	*	5	9	4	2	2	1	0	0
43	1	1	2	4	1	1	2	3	4	2	1	6	1	4	4	8	3	9	4	1	2	9	3	3	2	1	0	0
44	*	2	*	2	2	3	*	2	2	*	2	4	3	2	*	16	2	2	2	6	*	16	2	2	1	1	1	1
45	1	*	1	*	1	6	2	*	1	1	1	12	6	*	4	16	1	3	1	6	1	16	5	5	2	2	2	2
46	*	1	*	1	1	3	*	1	1	10	1	12	3	1	*	16	1	6	1	3	10	16	5	5	4	4	4	2
47	1	2	2	4	2	6	2	6	4	5	2	4	6	4	2	4	6	9	4	6	5	9	8	6	1	1	0	0
48	*	*	*	4	*	2	*	*	4	5	*	3	2	4	*	16	*	18	4	2	5	18	4	2	2	2	2	2
49	1	1	1	2	1	*	1	3	2	5	1	6	1	2	1	8	3	3	2	1	5	8	4	2	1	0	0	0
50	*	2	*	*	2	1	*	6	*	10	2	12	1	2	*	2	6	6	*	2	10	12	6	6	3	3	3	1
51	1	*	2	1	1	3	2	*	1	10	2	2	3	1	4	*	1	18	2	3	10	18	3	3	3	3	3	1
52	*	1	*	4	1	6	*	3	4	10	1	*	6	4	*	1	3	18	4	6	10	18	6	6	3	3	3	1
53	1	2	1	4	2	3	2	2	4	5	2	1	3	4	4	8	2	18	4	6	5	18	5	3	2	1	1	1
54	*	*	*	2	*	6	*	*	2	2	*	12	6	2	*	16	*	9	2	6	2	16	6	6	3	3	2	2
55	1	1	2	*	1	2	2	1	1	*	1	3	2	1	2	4	1	9	2	2	1	9	1	1	1	1	0	0
56	*	2	*	1	2	*	*	6	1	1	2	6	*	2	*	16	6	18	1	2	1	18	5	5	2	2	2	2
57	1	*	1	4	1	1	1	*	4	10	1	12	1	4	2	16	1	*	4	1	10	16	4	4	4	4	4	2
58	*	1	*	4	1	3	*	3	4	5	1	12	3	4	*	16	1	9	4	3	5	16	5	3	3	3	2	2
59	1	2	2	2	2	6	2	6	2	5	2	12	6	2	4	8	6	18	2	6	5	18	10	8	3	2	2	2
60	*	*	*	*	*	3	*	*	*	5	*	4	3	*	*	8	*	18	*	3	5	18	4	2	2	1	1	1
61	1	1	1	1	1	6	2	3	1	10	1	3	6	1	4	16	3	9	1	6	10	16	7	7	4	4	3	1
62	*	2	*	4	2	2	*	2	4	10	2	6	2	4	*	16	2	9	4	2	10	16	5	5	4	4	3	1
63	1	*	2	4	1	*	2	*	4	10	2	12	1	4	2	16	1	18	4	*	10	18	5	5	5	5	5	3

In recent years I have had considerable interest in comparing various bases regarding certain features of representations of various fractions. Unless the denominator contains no prime factor which does not divide the base (in which case the fraction will terminate) the representation of a fraction in any base will be recurring with a period equal to the smallest repunit divisible by the denominator in question.

A fraction with denominator n can have a period of at most n-1 regardless of base, and a period of n-1 is possible only for prime numbers. Such prime numbers are referred to as full period primes or full repetend primes, and except for perfect powers they generally constitute about three-eighths of all primes.

In the table above, I have attempted to compare the periods of reciprocals of all numbers from 2 to 22 in all bases from 2 to 63 — although for bases beyond 35 it is very difficult to write them out in easily understood notation. The aim is to see:

1. what base has the shortest maximum period for these reciprocals
2. what bases have the highest and lowest frequencies of long periods
• usually, a terminating expansion is taken as having period zero — effectively the shortest possible period

In the table above, I have compared:

1. the longest period in each base for the reciprocals of any number from 2 to 22
2. the number of numbers in that range with periods longer than 4, 5, 6, 8, 9 or 10
• no number smaller than 29 can have period 7 in any base, so the column “>7” was omitted after I experimented

Results:

The table above shows that for bases form 2 to 63:

• the shortest maximum period of reciprocal up to 22 is 6 for base 38, followed by 8 for bases 26 and 49 (all shaded dark green)
• the shortest maximum period of reciprocal up to 18 is 4 for bases 21, 34, and 55 (shaded light green)
• it is interesting to note that 21+34 = 55, and one wonders if there is a pattern involved?
• base 60, with all reciprocals up to 18 having periods of 5 or shorter, has also been shaded light green
• the most reciprocals up to 22 with periods longer than 4 is ten, for base 59
• the fewest reciprocals up to 22 with periods longer than 4 is one [in all cases the reciprocal of 19] for bases 21, 34 and 55

To more accurately consider the effect of base structure on period of reciprocal, I have compared bases by dividing them into five groups based upon factorisation:

1. prime bases
2. semiprime bases
3. nonsquare semiprime bases
1. I did this because square bases do not normally have any full period primes at all
2. thus, their maximum possible period is only half that of nonsquare bases
4. bases other than those in 1) or 2)
5. group 4) excluding square numbers

Comparative Results for Different Categories of Bases b
	Maximum	#>4	#>5	#>6	#>8	#>9	#>10
Prime	15.41176	6.352941	5.529412	2.882353	2.588235	2.235294	1.294118
Semiprime	14	4.045455	3.227273	2.272727	2.045455	1.681818	1.045455
Squarefree semiprime	15.16667	4.055556	3.5	2.444444	2.333333	2.055556	1.277778
Others	14.61905	4.761905	4.095238	2.428571	2.190476	1.952381	1.190476
Other nonsquare	15.21053	4.842105	4.315789	2.684211	2.473684	2.263158	1.315789

What appears to be the case is the prime bases have the greatest number of long periods, but semiprime bases on the whole have marginally fewer than bases with three or more factors. The difference between the three groups, though, is not large — indeed the prime base 47 is the smallest base where all reciprocals up to 24 have periods of nine or shorter. The one exception is that prime bases appear to have substantially fewer denominators with periods of 4 or shorter.

Further research could allow for investigation into questions like:

• how hard is it to find a base yielding consistently short periods for denominators of increasing size?
• how many such bases are there?
• can one calculate the smallest base for which all fractions up to n have periods of p or shorter with n and p arbitrary?
• what are the trends in frequency of denominators with short periods relative to size and number of factors in the base b as bases get bigger than studied here?