Crafting grafting, part 3

In the first two parts we found some grafting numbers using the equation $(j+x)^2=10x$ with $j=1~{\mathrm or~}2$. Real solutions $0 < x < 1$ can be turned into integer grafting number candidates $\lceil 10^{2n+1}x\rceil$. Or potentially $\lfloor 10^{2n+1}x\rfloor$ but we haven’t found a case where that works.

We could use other values for the coefficient of $x$, 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 $(j+x)^2=10^mx$. Solutions are $x = ((10^m-2j)\pm\sqrt{10^{2m}-4\times 10^mj})/2$. Real solutions are obtained up to $j=10^m/4$ but we want $0 < x < 1$. You can figure out this means $j < \sqrt{10^m}-1$.

For instance, with $m=2$, we can use $0 < j < 9$. With $j=1$, $x = 0.0102051443...$. Now $\lceil 10^{2n}(100x)\rceil$ = 2 ($n=0$), 103, 10206, 1020515, 102051444… not one of which, sorry to report, is a grafting number. With $j=8$, though, and $n = 0$, we get $\sqrt{77} = 8.77496...$ and that is a grafting number, though not very impressive. $j=6$ and $n = 4$ gives us $\sqrt{4110105646} = 64110.105646457...$. Yes! That’s what I’m talking about! And $j=7$ with $n = 1, 3, 5$ gives $\sqrt{5736} = 75.73638...$, $\sqrt{57359313} = 7573.5931366...$, and $\sqrt{57359312880715} = 7573593.1288071582...$.

And on it goes. Those two I started off the first post with, 60,755,907 and 63,826,090,875, arise from $j=7794$, $m=8$, and $n=0$ and $j=252$, $m=5$, and $n=3$, respectively. Here’s another: $\sqrt{44144658239614} = 6644144.658239614216...$, 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 $j=66$, $m=4$, and $n=5$. Then $x = 0.4414465823961399708...$, and $10^{14}x = 44144658239614$ 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: $j=2$, $m=1$, and $n=1$ giving grafting numbers 764 and 765. Otherwise it’s always ceil. Which suggests other questions to ask. Sometime.