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.
rank prime digits who when comment 1 Phi(39855, - 10) 21248 c95 Nov 2020 Unique, ECPP 2 Phi(23749, - 10) 20160 c47 Apr 2014 Unique, ECPP 3 Phi(14943, - 100) 18688 c47 Mar 2014 Unique, ECPP 4 Phi(18827, 10) 18480 c47 May 2014 Unique, ECPP 5 Phi(26031, - 10) 17353 c47 Apr 2014 Unique, ECPP 6 Phi(2949, - 100000000) 15713 c47 May 2013 Unique, ECPP 7 Phi(5015, - 10000) 14848 c47 Apr 2013 Unique, ECPP 8 Phi(13285, - 10) 10625 c47 Dec 2012 Unique, ECPP 9 Phi(427, - 10^{28}) 10081 FE9 May 2009 Unique, ECPP 10 Phi(5161, - 100) 9505 c47 Dec 2012 Unique, ECPP 11 Phi(6105, - 1000) 8641 c47 Jan 2010 Unique, ECPP 12 Phi(4667, - 100) 8593 c47 Dec 2009 Unique, ECPP 13 Phi(4029, - 1000) 7488 c47 Aug 2009 Unique, ECPP 14 Phi(9455, - 10) 7200 c33 Jul 2005 Unique, ECPP 15 Phi(1479, - 100000000) 7168 c47 Oct 2009 Unique, ECPP 16 Phi(2405, - 10000) 6912 c47 Apr 2009 Unique, ECPP 17 Phi(10887, 10) 6841 c33 May 2005 Unique, ECPP 18 Phi(7357, - 10) 6301 c33 May 2004 Unique, ECPP 19 Phi(6437, 10) 6240 c47 Nov 2008 Unique, ECPP 20 Phi(6685, - 10) 4560 c8 Mar 2003 Unique, ECPP
- 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.