Definition:Grafting Number

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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):

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

${\displaystyle 3-{\sqrt {5}}}$ 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):

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

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

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

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

Historical Note

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