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.


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