In the first two parts we found some grafting numbers using the equation with . Real solutions can be turned into integer grafting number candidates . Or potentially but we haven’t found a case where that works.
We could use other values for the coefficient of , though. Any power of 10 in fact. (For grafting numbers in base 10. If you want grafting numbers in another base, use powers of that base.) We have . Solutions are . Real solutions are obtained up to but we want . You can figure out this means .
For instance, with , we can use . With , . Now = 2 (), 103, 10206, 1020515, 102051444… not one of which, sorry to report, is a grafting number. With , though, and , we get and that is a grafting number, though not very impressive. and gives us . Yes! That’s what I’m talking about! And with gives , , and .
And on it goes. Those two I started off the first post with, 60,755,907 and 63,826,090,875, arise from , , and and , , and , respectively. Here’s another: ,
and that’s the only one I’ve found so far using floor instead of ceil. Edit: This does not use floor; I must have been fooled by a rounding error. This comes from , , and . Then , and when rounded up. In fact I’ve found no cases where floor gives the grafting number, and exactly one case where both ceil and 1+ceil work: , , and giving grafting numbers 764 and 765. Otherwise it’s always ceil. Which suggests other questions to ask. Sometime.