Definition:Grafting Number

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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:

$n$ $\sqrt n$ $n$ $\sqrt n$ 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):

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

$3 - \sqrt 5$ 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 $p = 2, b = 10, a = 1, c = 2$.



Historical Note

The concept of a grafting number was introduced by Matthew Thomas Parker.


Sources