User:HenryClay52/sandbox

Sleno's Number, or Ms. Sleno's Number, is the term for a mathematical constant discovered and studied locally by two high school students, Dyllon Denton and Kaiden Randol, which they found based on a problem made by their pre-calculus teacher, Ms. Sleno. It has since become a number of legend locally within Flushing High School lore. It is often denoted by s or sL. It is equivalent to 1.411522633744855967078189300, with the numbers after the decimal place repeating. The finding of the number, similar numbers to it, and the entire research process have since become legendary within the school.

History
The number was first discovered between 8 and 9 AM on December 9th, 2022, in the pre-calculus classroom of Flushing High School during first hour. It had come up while the class was practicing logarithms. The teacher, Ms. Sleno, came up with a problem to give the class some practice. The logarithm was:

$$2^{3log_2{7} - 5log_2{3}}$$

which condenses into

$$\frac{7^3}{3^5}$$

which calculates out to Sleno's Number. Therefore,

$$s_L = \frac{7^3}{3^5} = 1.411522633744855967078189300...$$

At first, no one thought anything of the number that appeared in their calculator. Upon further inspection, however, it was Kaiden Randol, known as one of the founders of the number and a discoverer of the Kaiden Sequence (which will be discussed later) who noticed a certain pattern with the first few digits past the decimal point. 411522633... It was not only a nice amount of repetition within the numbers, but the incremental increases in the value of the numbers made it seem special. Dyllon Denton, also known as one of the founders of Sleno's Number, looked into a precision calculator online which allowed them to calculate all 27 repeating digits of Sleno's Number. This was the birth of something amazing. Dyllon and Kaiden spoke up to Ms. Sleno about the fascinating pattern of the number, which Sleno promptly dubbed "Sleno's Number".

At this point, Dyllon and Kaiden took different approaches to their research throughout the day. Dyllon immediately started trying to calculate numbers that would produce the same pattern using algebra, and tried to figure out why $$\frac{7^3}{3^5}$$ worked so well. It turns out that 73 was 100 more than 35, meaning that the fraction simply became $$1 + \frac{100}{3^5}$$. This is when it was discovered that $$\frac{1}{3^5} = 0.004115226337448559670781893...$$, with the digits after the decimal point repeating. When multiplied by 100 and added to 1, this number calculates to Sleno's Number. Kaiden helped Dyllon immensely in learning this.

Shortly after this, Dyllon made a computer program to help him find similar values to Sleno's Number, though his methods and goals were flawed immensely. The program worked, but did little to quench Dyllon and Kaiden's thirst for knowledge on Sleno's number. Meanwhile, Kaiden had attempted to plug numbers into an equation, $$\frac{n^3}{3^5}$$, with n being a positive prime number. He found that all primes he plugged in for n created the same exact sequence, but at different points. For example, $$\frac{13^3}{3^5} = 9.041152263374485596707818930...$$, with every digit past the decimal point repeating. This led to an immense amount of discoveries by Dyllon, who theorizes that the equation, for any value of n that is not a multiple of 3, produces the same series of digits after the decimal point, just starting at different spots. Thus, the equation has been named the Randol-Denton Equation. Since then, Dyllon and Kaiden went on to find many more observations about the Randol-Denton Equation.

Findings

 * Sleno Sequence - The sequence of numbers following the decimal point of $$\frac{1}{3^5}$$, i.e., $$004115226337448559670781893$$. These digits repeat infinitely.
 * Kaiden Sequence - The sequence of numbers following the decimal point of $$\frac{2^3}{3^5}$$, i.e, $$032921810699588477366255144$$. These digits repeat infinitely.
 * Kaiden Function - The Kaiden function is often denoted as K(n). The function follows the following equation: $$K(n) = \frac{n^3}{3^5}$$, where n is a positive integer that is not a multiple of 3. Therefore, the output of the function is: $$0.004115226337448559670781893..., 0.032921810699588477366255144..., 0.263374485596707818930041152...$$, etc., where each number's digits after the decimal point repeat. This can also be denoted as $$\frac{1}{3^5}, \frac{2^3}{3^5}, \frac{4^3}{3^5}, \frac{5^3}{3^5}...$$ Therefore, another way to express Sleno's Number is as K(7).
 * Denton Diagram - The Denton Diagram is a diagram that shows the various starting points on the Sleno and Kaiden sequences when put into the Randol-Denton Equation. It also shows that when a number has a starting point on the Sleno Sequence, double that number will have a starting point on the same spot on the Kaiden Sequence (Ex, K(7) starts after the first digit of the Sleno Sequence, so K(14) will start after the first digit of the Kaiden Sequence). It also shows that each number 1-81 that is not a multiple of 3 has a unique starting point on the Sleno or Kaiden Sequences. After 81, the numbers loop. Therefore, 82 has the same starting point as 1, 83 as 2, etc. It loops after every multiple of 81. This means that if a number on the Sleno Sequence is doubled, and the doubled value is greater than 81, it must first be subtracted by 81 until it is less than 81; then it will be at the same starting point on the Kaiden Sequence. When doubling a number on the Kaiden Sequence, the Denton Diagram shows that the doubled value will be on the Sleno Sequence 7 starting points to the right (looping all the way to left if you run out of points on the right side).
 * First Denton Conjecture - The First Denton Conjecture is that any positive integer n that is not divisible by 3, when put into the Randol-Denton Equation, will always produce a number whose digits after the decimal point all follow the Sleno Sequence or Kaiden Sequence, just at different starting places. This is backed up graphically by the Denton Diagram.