Generalized Unique
The Prime Pages keeps a list of the 5000 largest known primes, plus a few each of certain selected archivable forms and classes. These forms are defined in this collection's home page.
This page is about one of those forms.
Definitions and Notes
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.
- Theorem.
- The prime p is a unique prime of period n
if and only if
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.
Record Primes of this Type
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
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.