Global maximum of ?(x)-x : guess maybe incorrect

[Gerald McGarvey]

It looks like my guess might be incorrect, for example,
letting c be the conjectured constant, c - 1/10^90 appears to have a
greater value of ?(x)-x, unless this is due to calculation error.

Of possible interest: local minimums at around
.31530096879 w/ CF [0, 3, 5, 1, 4, 1, 4, 1, 4, 1, 4, 1, 5, 1, 2, 2, 1, 1, 
1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 9, 4, 2]
and around
.69308651 w/ CF [0, 1, 2, 3, 1, 6, 1, 4, 1, 4, 1, 3, 1, 1, 1, 1, 1, 1, 2, 
1, 68, 2]

- Gerald

At 07:55 PM 6/13/2006, Gerald McGarvey wrote:
>I realized that narrowing the search for the maximum of ?(x)-x
>can be indeed be programmed.
>
>In a subinterval [a,b] of an interval being searched
>an upper bound for ?(x)-x is given by ?(b)-a
>since ?(x) is strictly increasing,
>so ?(b)-a can be compared with the maximum ?(x)-x value
>found to eliminate intervals and thus narrow the search
>to a smaller interval.
>
>The same idea can be applied to [](x)-x.
>
>I would be surprised if my guess at the maximum turns out to be correct.
>
>- Gerald
>
>At 04:52 AM 6/13/2006, Joseph Biberstine wrote:
>>Gerald McGarvey wrote:
>>>Based on a laborious semi-manual process (not recommended) of
>>>narrowing down the maximum, I believe the maximum of ?(x)-x
>>>occurs at a constant c whose continued fraction begins
>>>[0; 1, 3, 1, 4, 1, 4, 1, 5, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 5, 1, 4, 1, 
>>>4, 1, 4, 1, 4, 1, 4, 1, 5, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 5, 1, 4, 1, 
>>>4, 1, 4, 1, 4, 1, 4, 1, 5, 1, 4, 1, 4, 1, 4, 1, 4, 1, ...
>>>I hope I didn't miss a larger ?(x)-x along the way.
>>>Here's a conjecture: the maximum occurs at a constant c whose continued 
>>>fraction
>>>starts as [0; 1, 3, 1, 4, 1, 4, 1, 5, then has the repeating sequence 1, 
>>>4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 5
>>>so c = 1/(1+1/(3+1/(1+1/(4+1/(1+1/(4+1/(1+1/(5+1/((17325 + 
>>>3*sqrt(59115595))/33461)))))))))
>>>or (2343*sqrt(59115595) + 18014599)/(2955*sqrt(59115595) + 22720034)
>>>or
>>>.79289414860601842644318261515336048837629147616654626681193944753721958311485553800434172526654160238532789219294579... 
>>>
>>>If this is correct, the value of ?(c) appears to be 2008938429 / 2147483647.
>>>Of course, it could very well be incorrect.
>>>Has it be proven that ?(x) + ?(1-x) = 1 for all x?
>>>- Gerry
>><snip>
>>Can't say why, but this doesn't feel right to me.  I would have expected 
>>a monotonic increase at even entries.  The repetition also seems 
>>artificial.  It is surely a good approximation, but I expected a 
>>non-quadratic irrational for c.