(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?