Number factorization and a highly suspicous claim

[Gottfried Helms]

Am 17.10.05 02:33 schrieb Creighton Dement:
(citing a previous post)
> For x = -2'i -'j -'k,  the sequence 
> (tes[(Fx+Gx)], tes[(Fx+Gx)^2], tes[(Fx+Gx)^3], tes[(Fx+Gx)^4], ) "looks
> a lot like" two times the sequence
> http://www.research.att.com/projects/OEIS?Anum=A001333
> Numerators of continued fraction convergents to sqrt(2)
> 

Hi -

 I took a quick look at this convergents; a nice recurrence:

numerators of cf of sqrt(2)
Index                  factors of numerator  numerator  recurrence-def.
  0                                        1   1          2*-1+ 3
  1                                        1   1          2*1 - 1
  2                                        3   3          2*1 + 1
  3                                        7   7          2*3 + 1
  4                                       17   17         2*7 + 3
  5                                       41   41         2*17+ 7
  6                                 3^2 * 11   99         2*41+17
  7                                      239   239        2*99+41
  8                                      577   577
  9                                  7 * 199   1393
 10                              3 * 19 * 59   3363
 11                                 23 * 353   8119
 12                                17 * 1153   19601

A(i) = 2*A(i-1) + A(i-2)

Consequently this series shares the property of linear progressions
of indexpositions of its primefactors with the factors of lucas-
sequences and also is interesting in the focus of Szigmondy-factors
(only the first occurence of a factor is counted); the factors seem
either +1 or -1 modulo their index-position.


factor->Startindex(linear progression)
3->2(4)
Indexpositions:           factor of numerator  numerator of cf of sqrt(2)
  2                                        3   3
  6                                 3^2 * 11   99
 10                              3 * 19 * 59   3363
 14                            3 * 113 * 337   114243
 18                      3^3 * 11 * 73 * 179   3880899


7->3(6)
  3                                        7   7
  9                                  7 * 199   1393
 15                            7 * 31^2 * 41   275807
 21                         7^2 * 239 * 4663   54608393

17->4(8)
  4                                       17   17
 12                                17 * 1153   19601
 20                          17 * 241 * 5521   22619537
 28                          17 * 1535466241   2_61029_26097

41->5(10)
  5                                       41   41
 15                            7 * 31^2 * 41   275807
 25                            41 * 45245801   1855077841

11->6(12)
  6                                 3^2 * 11   99
 18                      3^3 * 11 * 73 * 179   3880899

Nice :-)



Gottfried Helms