Definition:Grafting Number

Definition
A grafting number of order $p$ 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 $2$nd order base $10$ grafting numbers.

Integers with this grafting property are called grafting integers (GIs).

For example, $98$ is a GI because:


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

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

$\begin{array}{r|l} n & \sqrt n \\ \hline 0 & \color {red} 0 \\ 1 & \color {red} 1 \\ 8 & 2. \color {red} {8} 28427 \dots \\ 77 & 8. \color {red} {77} 4964 \dots \\ 98 & \color {red} {9.8} 99495 \dots \\ 99 & \color {red} {9.9} 49874 \dots \\ 100 & \color {red} {10.0} \\ 764 & 2 \color {red} {7.64} 05499 \dots \\ 765 & 2 \color {red} {7.65} 86334 \dots \\ 5711 & 75. \color {red} {5711} 585 \dots \\ 5736 & 7 \color {red} {5.736} 3849 \dots \\ 9797 & 98. \color {red} {9797} 959 \dots \\ 9998 & \color {red} {99.98} 99995 \dots \\ 9999 & \color {red} {99.99} 49999 \dots \end{array}$

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):

$(1): \quad \ceiling {\paren {3 - \sqrt 5} \cdot 10^{2 n - 1} }, n \ge 1$

$3 - \sqrt 5$ 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):

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

$a, b, c, p$ are integer parameters where $p \ge 2$ is the grafting root, $b \ge 2$ is the base in which the numbers are represented, $a \ge 0$ is the amount the decimal point is shifted, and $c \ge 0$ 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 $p = 2, b = 10, a = 1, c = 2$.