Crafting grafting

Here’s a cool integer: 60,755,907. What’s cool about it? Take the square root. You’ll need more significant figures than usual; the calculator that came with my Android phone will do it:

\displaystyle \sqrt{60755907} = 7794.6075590756973...

Check out what’s to the right of the decimal point. Crazy, right? Here’s another similar number:

\displaystyle \sqrt{63826090875} = 252638.2609087546739...

Here the digits of the integer on the left appear in the real number on the right starting in the 100s place. Matt Parker calls these things “grafting numbers“. What’s going on with them? This isn’t just weird coincidence, is it?

It isn’t.

Consider good old \phi = (1+\sqrt{5})/2. It’s a solution of the equation x^2=x+1. So \phi = 1.6180339... and \phi^2 = 2.6180339.... There a number and its square (or, looked at the other way, a number and its square root) have digits in common; an infinite number of them, in fact. There’s a hint here.

Now let’s look at solutions to (j+x)^2 = 10x. Suppose 0 < x < 1. Now if j is an integer, then j+x in decimal form is just the concatenation of j and x and 10x is just the digits of x shifted one to the left.

For instance, j = 2; then (2+x)^2 = 10x. One solution is 3-\sqrt{5} = 0.7639320..., and 2.7639320...^2 = 7.639320.... Or looked at another way, \sqrt{7.639320...}=2.7639320....

This starts to look like the grafting numbers idea, but grafting numbers are integers. But hang on. Multiply both sides by, say, 100: \sqrt{76393.20...} = 276.39320.... Round the number on the left up and you find

\displaystyle \sqrt{76394} = 276.394645...

There you go, a grafting number. How about more? We’ll see…

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