Thursday, April 12, 2012

Experimental math notes: a curious prime sequence of length 22

The sequence is:
5, 3, 5, 3, 3, 11, 13, 13, 37, 17, 11, 17, 7, 5, 5, 17, 7, 17, 3, 17, 5, 31    (1)
An interesting property of this sequence: if P is the greatest prime factor function, then its infinite periodic extension (X(n)) satisfies the recursion
            X(n)=P[X(n-1)+3X(n-2)+2X(n-3)]    (2)
The intriguing part: for virtually all random choices of the initial conditions that I made - that is, other than the trivial ones (leading to 1-cycles, such as those with X(0)=X(1)=X(2)=p), the prime sequences satisfying (2) ultimately enter the same limit cycle (1). These are strange phenomena, of a similar nature to the GPF-tribonacci sequences (where we discovered that there are at least four distinct GPF-tribonacci limit cycles, of lengths 100, 212, 28 and 6). My last MAPLE experiment with (2) involved a seed consisting of three 80 digits primes,

X(0) = 67525204474446805798439049565857966823399463795779492274451732592404979647691781
X(1) = 43280708928363322606959208124229795876921456250899031972019611375562107511756173
X(2) = 50558523494317177773504742464927157574684415344640904005922668959550436919047547

which eventually led to the cyclic shift of (1) given by 7, 17, 3, 17, 5, 31, 5, 3, 5, 3, 3, 11, 13, 13, 37, 17, 11, 17, 7, 5, 5, 17, with 7=X(50), 17=X(51), 3=X(52), ...

  • Added on April 17, 2012 
Continued the search on possible other nontrivial limit cycles for the recursion X(n)=P[X(n-1)+3X(n-2)+2X(n-3)]. So far (1) is the only one found. Today - improved the process of random selection of initial conditions. A sample in the last series of searches - still leading to (1) (log plot of the graph included below)
X(0)=38790912184195861716665094754005872283807697220069306337918513393335402427949730921338400505241
X(1)=34380504590591201002245563
X(2)=239197136656882070249024044251770704653065407229190712881313724790604984709544507179663