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CONTINUED FRACTIONS


The continued fraction of log(N) and the powers of N
An interesting relation between powers of 2 and 3 was noted in L.E. Garner "On the Collatz 3n+1 algorithm", Proc. Am. Math. Soc. 82, 1981, 19-22.
Given the j-th power of 3, let p(j) be the largest power of 2 that is less than 3j. Therefore 2p(j) < 3j < 2p(j)+1 The author noticed that "the powers of 2 appear to be bounded away from the powers of 3 by an amount which grows almost as rapidly as the power of 3", and he defines

      b(m) = maxj ≤ m [-log(1 - 2p(j)/3j)]
      B(m) = maxj ≤ m [-log(2p(j)+1/3j - 1)]
Then the article presents a table of the values of m at which b(m) or B(m) increase. Here is the table with a few more columns:


  k      m     dm    p/P     dp     nr  b(m) B(m) 

  1______1______1______2______1______1___*____+_____
  2      2      1      3      2      1   +          
  3______3______1______5______2______2____ ___+_____
  4      5      2      8      3               +     
  5______7______2_____11______3______2___+__________
  6     12      5     19      8          +          
  7_____17______5_____27______8______3________+_____
  8     29     12     46     19               +     
  9     41     12     65     19               +     
 10_____53_____12_____84_____19______1___+__________
 11_____94_____41____149_____65______5________+_____
 12    147     53    233     84               +     
 13    200     53    317     84               +     
 14    253     53    401     84               +     
 15    306     53    485     84               +     
 16____359_____53____569_____84______2___+__________
 17    665    306   1054    485          +          
 18____971____306___1539____485_____23________+_____
 19   1636    665   2593   1054               +     
 20   2301    665   3647   1054               +     
 ...
 40  15601    665  24727   1054               +     
 41__16266____665__25781___1054______2___+__________
 42  31687  15601  50508  24727          +          
 ...


The rows can be grouped into blocks. A block ends after a shift from increasing b(m) to increasing B(m), or viceversa.
The number nr(m) are the numbers of rows in the block.


The quotients in bold p(m)/m, or P(m)/m, are "the best" approximations to log2(3).


The coefficients of the continued fraction expansion of log2(3) is the sequence A028507 in the On-line Encyclopedia of Integer Sequences.

1, 1, 1, 2, 2, 3, 1, 5, 2, 23, 2, 2, 1, 1, 55, 1, 4, 3, 1, 1, 15, 1,
9, 2, 5, 7, 1, 1, 4, 8, 1, 11, 1, 20, 2, 1, 10, 1, 4, 1, 1, 1, 1, 1,
37, 4, 55, 1, 1, 49, 1, 1, 1, 4, 1, 3, 2, 3, 3, 1, 5, 16, 2, 3, 1, 1,
1, 1, 1, 5, 2, 1, 2, 8, 7, 1, 1, 2, 1, 1, 3, 3, 1, 1, 1, 1, 5, 4, 2,
2, 2, 16, 8, 10, 1, 25, 2, ...


The convergents of this continued fraction are the "best approximations" fractions in the above table.



Prove that the numbers nr(j) are the terms of the sequence A028507.


Hint.
This is a general fact. It holds for any pair of numbers, not just 2 and 3.
Consider the differences |p(m)/m - log2(3)|.



Marco Corvi - 2016