## Great News About My Grafting Numbers Paper

A paper I wrote this past Summer on Grafting Numbers has been accepted for publication in JCMCC!

Here is the abstract of the paper:

Here is the abstract of the paper:

*Catalan Numbers and their generalizations are found throughout the field of Combinatorics. This paper describes their connection to numbers whose digits appear in its own p'th root. These are called Grafting Numbers and they are defined by a class of polynomials given by the Grafting Equation: (x+y)^p=x*b^a. A formula that solves for x in these polynomials uses a novel extension to Catalan Numbers and will be proved in this paper. This extension results in new sequences that also solve natural extensions to previous Combinatorics problems. In addition, this paper will present computationally verified conjectures about formulas and properties of other solutions to the Grafting Equation.*## Grafting Numbers

## Introduction

My interest in Grafting Numbers started when I watched the Numberphile video below, which features self-proclaimed Stand-Up Mathematician, Matt Parker, and discusses his discovery of Grafting Numbers. I highly recommend the rest of the Numberphile videos as well to any math enthusiasts out there.

Matt also discusses these grafting numbers in his blog. Being a computer programmer myself, I was intrigued by the way he used the computer program to find the numbers, then used math to find the patterns in the numbers, and then refined the computer program to find even more numbers.

As Matt describes, grafting numbers are numbers whose square root contains the number itself either before or directly after the decimal point. The root of the number grows off from the number itself, hence the name. I will refer to the integers such as 98, 99, 764, 5711, etc. as Grafting Integers (GIs) to differentiate them from the actual Grafting Numbers(GNs) such as the one shown in the video: 3

As Matt describes, grafting numbers are numbers whose square root contains the number itself either before or directly after the decimal point. The root of the number grows off from the number itself, hence the name. I will refer to the integers such as 98, 99, 764, 5711, etc. as Grafting Integers (GIs) to differentiate them from the actual Grafting Numbers(GNs) such as the one shown in the video: 3

**-**sqrt(5). A table of the first few GIs is shown below:## Finding Patterns in the Numbers

At first glance, other than the 99, 98, 9999, 9998 pattern described by Matt at the beginning of his video, these numbers look mostly random. However, A few more GIs in the pattern that includes 8 and 764 that Matt found are 76394, 7639321, 763932023, ... These digits correspond to our first discovered Grafting Number: 3

sqrt(10x) = x+2

10x = (x+2)^2

10x = x^2 + 4x + 4

0 = x^2

x = 3

So sqrt(7.63932023...) = 2.763932023... and, as you can see, all the digits on the left side are present on the right side! If the number inside the sqrt is multiplied by 100, then the number on the right side is multiplied by sqrt(100) = 10 which is why the pattern works every two digits (8, 764, 76394, etc.). Matt shows a formula which rounds up the GN to create GIs as shown in equation (1) below:

(1) ceiling[(3

The reason that the floor function doesn't work, and rounding up from, say, 7.63932023 to 8 still works is a more advanced topic involving how a small change to the number in the sqrt affects the result.

What about the other solution, 3+sqrt(5) = 5.23606798...? Does that create a GN as well? Well sqrt(52.3606798) = 7.23606798..., so it almost works, but because the number is greater than 1, adding 2 to it changes the first digit. So now we know that the smaller solution for x to the equation sqrt(10x) = x + 2 yields a Grafting Number. This equation can also be written as (x+2)^2 = 10x. What about (x+1)^2 = 10x? Well the solutions to that equation are 4+sqrt(15) and 4

**-**sqrt(5) = 0.763932022500210... The reason this works is because 3**-**sqrt(5) is a solution for x in the equation sqrt(10x) = x + 2 as shown below (you can play around with equations like this yourself here):sqrt(10x) = x+2

10x = (x+2)^2

10x = x^2 + 4x + 4

0 = x^2

**-**6x + 4x = 3

**-**sqrt(5) or 3+sqrt(5)So sqrt(7.63932023...) = 2.763932023... and, as you can see, all the digits on the left side are present on the right side! If the number inside the sqrt is multiplied by 100, then the number on the right side is multiplied by sqrt(100) = 10 which is why the pattern works every two digits (8, 764, 76394, etc.). Matt shows a formula which rounds up the GN to create GIs as shown in equation (1) below:

(1) ceiling[(3

**-**sqrt(5)) * 10^(2n**+**1)] for n = 0, 1, 2, 3, ...The reason that the floor function doesn't work, and rounding up from, say, 7.63932023 to 8 still works is a more advanced topic involving how a small change to the number in the sqrt affects the result.

What about the other solution, 3+sqrt(5) = 5.23606798...? Does that create a GN as well? Well sqrt(52.3606798) = 7.23606798..., so it almost works, but because the number is greater than 1, adding 2 to it changes the first digit. So now we know that the smaller solution for x to the equation sqrt(10x) = x + 2 yields a Grafting Number. This equation can also be written as (x+2)^2 = 10x. What about (x+1)^2 = 10x? Well the solutions to that equation are 4+sqrt(15) and 4

**-**sqrt(15), again, since the bigger solution is greater than 1, we will only be looking at the smaller solution, 4**-**sqrt(15) = 0.12701665379..., and this is our second discovered Grafting Number! Checking some possible GIs using a similar formula to equation (1) with this GN instead yields, sqrt(128) = 11.313708... and sqrt(12702) = 112.7031... The last digits are not quite right! This is why no numbers of this type show up in the GI chart above. The rounding doesn't quite work all the time. In fact, the first integer that actually works as a GI is sqrt(127016654) = 11270.166547128...