unique prime
The reciprocal of every prime p (other than two and five) has a period, that is the decimal expansion of 1/p repeats in blocks of some set length (see the period of a decimal expansion). This is called the period of the prime p. Samuel Yates defined a unique prime (or unique period prime) to be a prime which has a period that it shares with no other prime. For example: 3, 11, 37, and 101 are the only primes with periods one, two, three, and four respectively--so they are unique primes. But 41 and 271 both have period five, 7 and 13 both have period six, 239 and 4649 both have period seven, and each of 353, 449, 641, 1409, and 69857 have period thirty-two, showing that these primes are not unique primes.As we would expect from any object labeled "unique," unique primes are extremely rare. For example, even though there are over 10^{47} primes below 10^{50}, only eighteen of these primes are unique primes (all listed in the table below). We can find the unique primes using the following theorem.
- Theorem.
- The prime p is a unique prime of period n
if and only if
It is possible to generalize this to other bases, and the generalized unique primes in base-x (any integer greater than one) are the prime factors of which do not divide x.
period | prime |
---|---|
1 | 3 |
2 | 11 |
3 | 37 |
4 | 101 |
10 | 9091 |
12 | 9901 |
9 | 333667 |
14 | 909091 |
24 | 99990001 |
36 | 9999990000 01 |
48 | 9999999900 000001 |
38 | 9090909090 90909091 |
19 | 1111111111 111111111 |
23 | 1111111111 1111111111 111 |
39 | 9009009009 0099099099 0991 |
62 | 9090909090 9090909090 9090909091 |
120 | 1000099999 9989998999 9000000010 001 |
150 | 1000009999 9999989999 8999990000 0000010000 1 |
Related pages (outside of this work)
- The largest known unique primes
- The largest known generalized unique primes
References:
- Caldwell97
- C. Caldwell, "Unique (period) primes and the factorization of cyclotomic polynomial minus one," Mathematica Japonica, 46:1 (1997) 189--195. MR 99b:11139 (Abstract available)
- CD1998
- C. Caldwell and H. Dubner, "Unique period primes," J. Recreational Math., 29:1 (1998) 43--48.
- Yates1980
- S. Yates, "Periods of unique primes," Math. Mag., 53:5 (1980) 314.
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