(I’ve gone back and revised my notation. becomes
and
becomes
.)
Check out this grafting number, the second biggest I’ve got at the moment:
There’s some structure to understand here. This is generated using ,
, and
. You can read those values off the numbers above, though. There are 22 digits in the integer on the left, and that’s
. On the right, those 22 digits recur starting 7 digits left of the decimal point; that’s
. Before them are the digits
, which is
.
Likewise, examine this one:
You can see and
, so
, and
.
Though here’s an evil one:
Here and
, so
, and
. Or… wait, is it? If you write the integer on the left as
then
and
so
and
. Or should you write it as
and conclude
and
so
and
? Turns out all three interpretations are valid; they lead to three different equations yielding the same grafting number.
What about this miserable attempt at a grafting number?
Well, and
, so
, um,
? And
? Those give
(with the
sign in the quadratic formula, unlike all normal grafting numbers, because the
sign gives
) and the equation
or
. Yes, that sort of fits the profile, but in an ugly way.
Likewise, and
have similarly pathological analyses. Those I think really do need to be counted as grafting numbers, but not normal ones.
Two things I have not figured out yet is why it’s always the ceiling, not the floor, that gives the grafting number, and why and
gives grafting numbers for, apparently, all values of
while other values of
and
don’t. I suspect these are related. Observe
(with braces denoting fractional part) which is, to first order
and is whatever it is, but it’s in
and for an approximate result we can take
so
.
By taking the square root of the ceiling of a number instead of
itself, we add approximately
to the square root. By the same token, using floor subtracts about
.
That’s sort of an average, though, and in the case of grafting numbers, it’s often an overestimate. For instance, for ,
, we get
. Then with
we end up with grafting number
. Clearly the ceiling in this case changes the number by several orders of magnitude less than
. But why don’t numbers looking hypothetically like
work out?