Global maximum of ?(x)-x

[Gerald McGarvey]

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.