Generalized Unique |
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. We can find the unique primes using the following theorem.
is a power of p where is the nth cyclotomic polynomial.
It is possible to generalize this to other bases: A prime p is a unique prime in base b, if and only if, for some integer n, it is the only prime divisor of Φ_{n}(b) which does not divide n (and it may occur as a prime power). For example, the Gaussian Mersenne norms (other than 5) all have the form
2^{n} ± 2^{(n+1)/2} +1.Since Φ_{4}(x) = x^{2}+1, these can be written as Φ_{4}(2^{(n+1)/2}± 1)/2, showing these are all generalized unique primes.
>rank prime digits who when comment 1 Phi(3, - 123447^{524288}) 5338805 L4561 Feb 2017 Generalized unique 2 2^{15317227} + 2^{7658614} + 1 4610945 L5123 Jul 2020 Gaussian Mersenne norm 41?, generalized unique 3 Phi(3, - 143332^{393216}) 4055114 L4506 Jan 2017 Generalized unique 4 Phi(3, - 844833^{262144}) 3107335 L4506 Jan 2017 Generalized unique 5 Phi(3, - 712012^{262144}) 3068389 L4506 Jan 2017 Generalized unique 6 Phi(3, - 558640^{196608}) 2259865 L4506 Jan 2017 Generalized unique 7 Phi(3, - 237804^{196608}) 2114016 L4506 Jan 2017 Generalized unique 8 Phi(3, - 1082083^{131072}) 1581846 L4506 Jan 2017 Generalized unique 9 Phi(3, - 843575^{131072}) 1553498 L4506 Jan 2017 Generalized unique 10 Phi(3, - 362978^{131072}) 1457490 p379 May 2015 Generalized unique 11 2^{4792057} - 2^{2396029} + 1 1442553 L3839 Apr 2014 Gaussian Mersenne norm 40?, generalized unique 12 Phi(3, - 192098^{131072}) 1385044 p379 Feb 2015 Generalized unique 13 Phi(3, - 1202113^{98304}) 1195366 L4506 Dec 2016 Generalized unique 14 Phi(3, - 1110815^{98304}) 1188622 L4506 Dec 2016 Generalized unique 15 Phi(3, - 700219^{98304}) 1149220 L4506 Dec 2016 Generalized unique 16 Phi(3, - 660955^{98304}) 1144293 L4506 Dec 2016 Generalized unique 17 Phi(3, - 535386^{98304}) 1126302 L4506 Dec 2016 Generalized unique 18 2^{3704053} + 2^{1852027} + 1 1115032 L3839 Sep 2014 Gaussian Mersenne norm 39?, generalized unique 19 Phi(3, - 406515^{98304}) 1102790 L4506 Dec 2016 Generalized unique 20 Phi(3, 3^{1118781} + 1)/3 1067588 L3839 Mar 2014 Generalized unique
- 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.