log(m)'s close to integers, various log bases

[Neil Fernandez]

In message <LW9ntHj034n at paradise.net.nz>, Don McDonald
<parabola at paradise.net.nz> writes
>In message <sCLBGvAnbNL+EwtX at borve.demon.co.uk> you write:
>
>> I wonder whether phi is unique in that every one of its powers has, as
>> the nearest integer to it, a term in its sequence; and no term in the
>> sequence is not the nearest integer to one of its powers.

>> Neil Fernandez
>>
>I presume phi = golden ratio = (sqrt(5)+1  ) / 2.
>
>In which case it is trivial to calculate exact powers of a surd.
>No, I think it is not unique?

Apologies if I am missing something, or have misunderstood something,
but have you got an example of a positive number x other than phi that
meets the criteria? I was keeping to Leroy's definition:

>>>>>Consider the integers m, m >= 2, such that:
>>>>>ln(m) is closer to an integer than any previous term of the
>>>>>sequence

but generalising to allow the log base to be x.

Even if we allow x<1 and therefore get negative integers for log(m),
there is no x<1 that meets the first criterion. This is assuming that we
begin counting powers from x^1. The nearest integer to x^1 will be
either 0 or 1 but neither of these numbers will be in the sequence.
(It's of interest though that x=(phi-1) yields the same sequence as for
x=phi).

The sequence for x=sqrt(2) is 2 (stops)
The sequence for x=sqrt(5) is 2, 5 (stops)
(finite sequences fail to meet the first criterion)

The sequence for x=sqrt(2)+1 is 5, 6, 14, 34, 82, 198, 478,...
          (x^2 has two corresponding terms, namely 5 and 6;
           5 is not the closest integer to a power of x)
The sequence for x=sqrt(5)+1 is 2, 3, 10, 33, 34, 110, 354, 355,...
          (2, 33, and 354 are not the closest integers to a power of x)
(these fail to meet the second criterion)
In fact all numbers x>2.5 give sequences that include 2, and since this
is not the nearest integer to a power of x, these numbers fail to meet
the second criterion.

If there is another number than phi that works (I conjecture that there
isn't), then it is greater than 1 and smaller than 2.5.

Neil
-- 
Neil Fernandez