This thread concerns itself with the use of Wieferich primes w to determine factors in ratios merely by their base b expansions.
(See this earlier thread on sevenites 12 September 2012 for further explanations of Wendy's term.)
Those interested in the mathematical definition of a Wieferich prime can educate themselves here, with a "friendlier" explanation here. Of interest to us are base b Wieferich primes 2 <= w <= lim, the limit being some reasonably small prime that might be useful to us. The smallest w for base b is given by OEIS A039951.
The practical application in base b of w and certain small integer powers e of w is the sharing of a recurrent period, i.e., the length of the group of digits after the radix point that is repeated. All Wieferich primes w are coprime to b thus they are purely recurrent as denominators.
Let's look at 1/5 and 1/25 in base 18. The former is ".3a37..." and the latter is ".0ch5...". Now 1/5 is maximally recurrent (which is seen as a disadvantage); this aside, we see that 1/25 is no worse in terms of period. (The base-b expansion of the reciprocal of a prime q coprime to b is maximally recurrent if its period is b - 1.) A better situation subtends for 1/7, 1/49, and 1/343, the smallest three positive integer powers of 7 in base 18: these are ".2a5...", ".06b...", and ".00h" respectively. These powers share the relatively brief period of 1/7 in base 18.
In octal, 3 is a Wieferich prime: 1/3 and 1/9 are ".25..." and ".07..." respectively. This makes working with thirds handier than it otherwise would be, despite the fact that 3 is coprime to 8.
Perhaps the very most stunning case of a Wieferich prime's utility occurs in decimal regarding w = 3. Decimally 1/3 and 1/9 are ".3..." and ".1..." respectively. The prime divisors of 10 are 2 and 5; with the minimally recurrent 3 supplying a minimally recurrent (by definition, as 3^2 = 9 = b - 1 or the decimal "omega"). Despite the fact that 10 has "skipped" the prime 3, decimal users enjoy some transparency of the divisibility of an arbitrary decimal integer by 3 and 3^2. Everyone is familiar with the decimal divisibility rules pertaining to 3 and 9. To determine if a number n is divisible by 3, simply total the digits of n. If the total is divisible by 3 then n is also divisible by 3. This divisibility rule is perhaps the simplest such rule outside of examining the number of zeros at the end of a number to determine if n is divisible by the base. This "infilling" of accommodation for the "skipped" factor 3 makes decimal more useful than it otherwise would be.
The following table gives Wieferich primes w with indices of 30 or below, i.e., primes {2, 3, ..., 113} for bases 2 through 120 inclusive:
Code: Select all
2:
3: 11
4:
5: 2
6:
7: 5
8: 3
9: 2, 11
10: 3
11: 71
12:
13: 2
14: 29
15:
16:
17: 2, 3
18: 5, 7, 37
19: 3, 7, 13, 43
20:
21: 2
22: 13
23: 13
24: 5
25: 2
26: 3, 5, 71
27: 11
28: 3, 19, 23
29: 2
30: 7
31: 7, 79
32: 5
33: 2
34:
35: 3
36:
37: 2, 3
38: 17
39:
40: 11, 17
41: 2, 29
42: 23
43: 5, 103
44: 3
45: 2
46: 3
47:
48: 7
49: 2, 5
50: 7
51: 5, 41
52:
53: 2, 3, 47, 59, 97
54: 19
55: 3
56:
57: 2, 5
58:
59:
60: 29
61: 2
62: 3, 19
63: 23, 29
64: 3
65: 2, 17
66:
67: 7, 47
68: 5, 7, 19, 113
69: 2, 19
70: 13
71: 3, 47
72:
73: 2, 3
74: 5
75: 17, 43
76: 5, 37
77: 2
78: 43
79: 7
80: 3, 7, 13
81: 2, 11
82: 3, 5
83:
84:
85: 2
86:
87:
88:
89: 2, 3, 13
90:
91: 3
92:
93: 2, 5
94: 11
95:
96: 109
97: 2, 7
98: 3
99: 5, 7, 13, 19, 83
100: 3
101: 2, 5
102:
103:
104:
105: 2
106:
107: 3, 5, 97
108:
109: 2, 3
110: 17
111:
112: 11
113: 2
114:
115: 31
116: 3, 7, 19, 47
117: 2, 7, 31, 37
118: 3, 5, 11, 23
119:
120: 11