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. Comments and suggestions requested.
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, and the generalized unique primes in base-x (any integer greater than one) are the prime factors of which do not divide x.
rank prime digits who when comment 1 Phi(3, - 123447^{524288}) 5338805 L4561 Feb 2017 Generalized unique 2 Phi(3, - 143332^{393216}) 4055114 L4506 Jan 2017 Generalized unique 3 Phi(3, - 844833^{262144}) 3107335 L4506 Jan 2017 Generalized unique 4 Phi(3, - 712012^{262144}) 3068389 L4506 Jan 2017 Generalized unique 5 Phi(3, - 558640^{196608}) 2259865 L4506 Jan 2017 Generalized unique 6 Phi(3, - 237804^{196608}) 2114016 L4506 Jan 2017 Generalized unique 7 Phi(3, - 1082083^{131072}) 1581846 L4506 Jan 2017 Generalized unique 8 Phi(3, - 843575^{131072}) 1553498 L4506 Jan 2017 Generalized unique 9 Phi(3, - 362978^{131072}) 1457490 p379 May 2015 Generalized unique 10 Phi(3, - 192098^{131072}) 1385044 p379 Feb 2015 Generalized unique 11 Phi(3, - 1202113^{98304}) 1195366 L4506 Dec 2016 Generalized unique 12 Phi(3, - 1110815^{98304}) 1188622 L4506 Dec 2016 Generalized unique 13 Phi(3, - 700219^{98304}) 1149220 L4506 Dec 2016 Generalized unique 14 Phi(3, - 660955^{98304}) 1144293 L4506 Dec 2016 Generalized unique 15 Phi(3, - 535386^{98304}) 1126302 L4506 Dec 2016 Generalized unique 16 Phi(3, - 406515^{98304}) 1102790 L4506 Dec 2016 Generalized unique 17 Phi(3, 3^{1118781} + 1)/3 1067588 L3839 Mar 2014 Generalized unique 18 Phi(3, - 107970^{98304}) 989588 L4506 Dec 2016 Generalized unique 19 Phi(3, - 62721^{98304}) 943210 L4506 Dec 2016 Generalized unique 20 Phi(3, - 1538654^{65536}) 810961 L4561 Oct 2017 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.