Relationship of nine constants

Through the process of trial and error, four unitless equations made up of nine constants have been found with exact answers. The related constants are the Speed of Light [1], the Planck constant [2], Wien’s displacement constant [3], Avogadro’s number [4], the universal Gravity constant [5], the Ampere constant [6], the Faraday constant [7], the Gas constant [8] and Apery’s constant [9].


INTRODUCTION
At the end of the spring semester 2013, I had found an expression of a few physical constants that gave the correct value of the universal Gravity constant [5].I shared my findings with my classmates and they all pointed out the units were incorrect.
This started the search for a unitless expression of physical constants similar in form to the Fine Structure constant but with more constants.
In the context of this paper, the term unitless is defined as all the exponents of the units on the left hand side of the equation are equal to zero and the right hand side of the equation is represented by only a numeric expression.

MAIN BODY
The first few equations were found by trial and error.One would literally examine a listing of physical constants and guess which set of constants multiplied together and divided by another set of multiplied constants produces an answer with units raised to the zero power.
I had the limited success of finding the Fine Structure constant over and over again.At this point I changed my strategy by writing a program that would try every combination of a set of constants within a certain integer range of exponents, with its dimensionality equal to a selected SI unit.This strategy worked in the sense that it produced a large set of equations, of the selected constants that had the required SI units of seconds or meters, etc.
The programming process and the testing the programs happened over a few weeks and various sets of physical constants where tried.Overall the physical constants that produced the most equations were selected to be in the final set of nine presented in this paper.
A few things happened concurrently that allowed me to find the equations presented in this paper.One was that I started using a unit of an ampere-mole as a range extender in my search programs.The derived unit could be removed from the final answers yet its presence in the program allowed more equations to be found.
The second was that it occurred to me that the structure of the programs that I had written; could search for unitless equations too.The third was that I added the Faraday constant to the primary set of search constants.I intended to use the Faraday constant as a more robust replacement for the derived ampere-mole constant, and was hoping for similar results.
A few minutes later, the first of many unitless equations appeared on the screen.Through the process of trial and error I had found a set of eight physical constants that produced unitless equations.
Once a pattern was found in the first few equations, a new program in the Cuda GPU language was written to find unitless combinations expressed as the powers of the constants.A program listing is included for completeness as Appendix I.
A set of 200 unitless equations are shown in Table 1, and Eq.1 through Eq.4 are the results of the reduced row echelon form of Table 1.The reduced row echelon operation on Table 1, results with two rows.
Eq.1 represents the first row and Eq.2 represents the second row of the reduced row echelon form.Eq.3 represents the multiplication of the Eq.1 and Eq.2 and Eq.4 represents the quotient of Eq.1 and Eq. 2. One can

DISCUSSION
Once we know that the dimensionality of the left hand sides of the equations are correct, then our focus switches to the right hand side of the equations.One should note by definition all the physical constants on the left hand side are measurements and have limited accuracy.
    Obviously the equations based physical measurements can not be more accurate than the measurements themselves.My method was to give the Maple software program the benefit of the doubt when computing the right hand sides of the equations.9    (4) For example while factoring and processing Eq.3 with Maple's identify command, the Apery's constant [9] appears in the result.Apery's constant can be expressed as a series, which means we could convert the right hand side of Eq.3 into a series just by redistributing some of the factors.For this reason I left Apery's constant in the answer which propagated to other the equations.
Figure 1 is a plot of Table 1, and is intended to show that the system of equations in Table 1 is not random but very periodic.The green line represents the natural log of the right hand sides of the equations and the other lines represent the exponent powers of the physical constants.
I view the form of the right hand sides of the equations as an idealized guess times an error term which was supplied by the reduced row echelon operation.
Figure 2 is also a plot of Table 1, where the equations of the table have been resorted based on the values of the ninth column of the table, instead of the tenth column.A problem is that the right hand side of the equations are inherently more accurate than the left hand side of the equations; which means any exact answer found by my method is merely a good guess.
On the other hand, these guesses appear to have over seven significant digits of accuracy.Practically speaking, the right hand sides of Eq.1 through Eq.4 are close enough to the "right answers" to solve most problems and if one wishes more accuracy one can always use the left hand side to directly compute a decimal value.

SUMMARY
In some ways, this paper is mundane.We have a family of similar equations where any single equation can be proven with dimensional analysis to be unitless.
Assuming that a suitable expression can be found for the right hand sides of the equations, then most of these equations could be used like a Swiss army knife to change from one physical constant to another.
On the mundane side we basically have a relationship between nine constants that connects the constants like a key ring.On the other hand, one could argue that the relationships shown in this paper existed before any of the physical constants were measured.
Obviously I can not address the range of philosophical issues that this paper may cause.To answer the reader's unspoken question, I do not know why these relationships exist; I only know that each time that I check them they seem to be correct.I invite other papers to address the deeper issues and physical interpretations of my equations.
A database of over 17,000 equations is available for download; the reader is encouraged to download the database and verify my work.By definition the terms of these equations tend to be self canceling, meaning if you make the wrong substitution, the whole left hand side can disappear and just leave a number.This has happened to me quite a few times in the last few months, which leads me to my final statement of the paper: "I claim nothing."

Figure 1 .
Figure 1.Plot ofTable 1 sorted by the right hand side values.

Table 1 .
A family of unitless equations.
Figures1 and 2should prove that the family of unitless equations contained in Table1is not random but instead is a structure made up of periodic waveforms.use dimensional analysis to check that Eq.1 through Eq.4 are unitless equations.

Table 1
sorted by the right hand side values.

Figure 2 .
Plot of Table 1 sorted by the exponent values of the Faraday constant term.