"Egyptian Continued Fractions"

[Hans Havermann]

> (1) What are the smallest partial quotients of the continued fraction
> whose fractional remainders add to unity?
>
> The continued fraction may begin like this (252238200 is approximate):
>     0.4042503503074... = [0;2,2,9,91,14201,252238200,...]
> so that
> 1 = [0;2,2,9,91,14201,...] + [0;2,9,91,14201,...] + [0;9,91,14201,...]  
> +
> [0;91,14201,...] + [0;14201,...] + ...
>   = .40425035 + .47371461 + .11097560 + .01098900 + .00007041 + ...


[0; 2, 2, 9, 91, 14201, 252238179, 82413709268226496,  
12393783734739289765092773334814410,  
940449499772176767594719706273493318801155215211368219531441729200804,  
101994455102164785576228857763408315426702687211933642117611818896456293 
87222768042679048998974524621602038845983259639811417376376669729813,  
...]