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A grafting number[1] is a number whose digits, represented in base b, appear before or directly after the decimal point of its p'th root.  The simplest type of grafting numbers, where b=10 and p=2, deal with square roots in base 10 and are referred to as 2nd order base 10 grafting numbers.  

Integers with this grafting property are called grafting integers (GIs).[2]  For example, 98 is a GI because:

$ \sqrt{98} = \mathbf{9.8}9949 \, $

The 2nd order base 10 GIs between 0 and 9999 are:

$ n $ $ \sqrt{n} $ $ n $ $ \sqrt{n} $
0 0 764 27.6405499...
1 1 765 27.6586334...
8 2.828427... 5711 75.5711585...
77 8.774964... 5736 75.7363849...
98 9.899495... 9797 98.9797959...
99 9.949874... 9998 99.9899995...
100 10.0 9999 99.9949999...


More GIs that illustrate an important pattern, in addition to 8 and 764, are: 76394, 7639321, 763932023, and 76393202251.  This sequence of digits corresponds to the digits in the following irrational number

$ 3 - \sqrt{5} = 0.76393202250021019... $

This family of GIs can be generated by Equation (1):

$ (1)\ \ \ \lceil (3 - \sqrt{5}) \cdot 10^{2n-1} \rceil, n \geq 1 $

$ 3 - \sqrt{5} $ is called a grafting number (GN), and is special because every integer generated by (1) is a GI. For other GNs, only a subset of the integers generated by similar equations to (1) produce GIs.

Each GN is a solution for $ x $ in the Grafting Equation (GE):

$ (GE)\ \ \ (x \cdot b^{a} )^{1/p} = x+c $

$ a, b, c, p $ are integer parameters where $ p\geq 2 $ is the grafting root, $ b\geq 2 $ is the base in which the numbers are represented, $ a\geq 0 $ is the amount the decimal point is shifted, and $ c\geq 0 $ is the constant added to the front of the result.

When $ 0 < x < 1 $, all digits of $ x $ represented in base $ b $ will appear on both sides of the Equation (GE).

For $ x=3 - \sqrt{5} $ the corresponding values are $ p = 2, b = 10, a = 1, c = 2 $.

References

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  1. http://www.numberphile.com/videos/98grafting.html
  2. http://roberttanniru.weebly.com/grafting-numbers.html

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