Math Is Fun Forum

My research concerns the Multiplication Table.

Everyone is probably familiar with the grade school multiplication table that is either 10 by 10 or 12 by 12.

Running at a 45 degree angle from the 1 times 1 position is a line of numbers that I call the square line. All of the numbers in that line are perfect squares in order of their occurrence. This line of numbers is infinite. And of course the entire multiplication table is infinite.

There are many patterns in the multiplication table. I have discovered at least one that I’m not sure is known.

If you construct a 90 by 90 multiplication table based on the way numbers end combined with their digital roots you will find 90 patterns of numbers. I have plotted all 90 patterns on Excel worksheets. I have also copied the worksheets to Word documents for analysis and comparison. This research along with my conclusions has taken a number of years. I will discuss the 90 patterns further on down.

Research into the multiplication table came about as a result of the age old search for the identification of prime numbers. On the multiplication table prime numbers can only be found on the top most row and on the left most column. Yet, there many numbers on the multiplication table that have two characteristics that prime numbers have.

Prime numbers have two characteristics that are important for this discussion. Except for the numbers 2 and 5, one characteristic of prime numbers is that they all end in 1, 3, 7 or 9. A second characteristic of prime numbers that is important for this discussion is that their digital roots are not 3, 6 or 9. Numbers with digital roots of 3, 6 or 9 are divisible by 3. So, the digital root of a prime number can only be 1, 2, 4, 5, 7 or 8.

Curiosity led to a search of the Multiplication Table. The object was to see where the numbers, that had the two characteristics described above, would fall out on the multiplication table.

The preliminary search indicated that there seemed to be some kind of pattern to the numbers that had these two characteristics.

It was quickly realized, though, that the numbers would become large so quickly that they wouldn’t fit within the individual cells on the table. So the decision was made to search the table using the digital roots of the numbers.

Working with a large sheet of graph paper an attempt was made to see where numbers which end in 1, 3, 7 or 9 with digital roots of 1, 2, 4, 5, 7 or 8, fall out on the multiplication table. After some trial and error and a few false starts, working with all of the endings and all of the digital roots, a new approach was devised.

To reduce confusion the new approach used only one ending and only one digital root. I selected Numbers ending in 1 with a Digital Root of 2. What prompted me to choose that particular combination of ending and digital root was that I needed a number that I knew would not be a prime number and that had the two characteristics being researched. The number 1001 fit the bill nicely. It is the product of 7 X 11 X 13.

Anyway, after working for a few hours I noticed that a pattern was emerging. When arbitrary lines were drawn between some of the points, a geometric pattern emerged that looked like a “Y”. More lines were drawn and revealed another shape that looked like an upside down “Y”. I cannot show it here, but the two “Y” patterns are orientated at a 45 degree angle and are in line with the so called square line.
After more of these Y’s were found I joked with my friend, “Well we might not solve the Prime Number Problem but we sure do have a nice wallpaper pattern.”

The first two Y’s made it easier to locate more of the positions of numbers that ended in 1 with a digital root of 2. And it soon became apparent that these Y’s would continue infinitely down through the multiplication table at the same 45 degree angle as is the Square Line.

Now the square line is also the demarcation line that divides the multiplication table into mirror images. All of the numbers that proceed outward to the right of the square line are the exact mirror image of all of the numbers that proceed downward from the square line. The square line itself is unique.

It came to notice also that the same “Y” pattern would repeat itself over and over again outward to the right of the square line and downward from the square line. And it was observed that the “Y” patterns that faced each other would repeat themselves over and over again.

Arbitrary lines were constructed dividing the repeating “Y” patterns into groups similar to the way the square line did. The first number on the first arbitrary line is 91. More studies further out on the table revealed that the next line would start at 181 and the line beyond that would start at 271.

An epiphany occurred. It was perceived that any 90 by 90 block of numbers that started with a multiple of 90 + 1 would have the same pattern as any other 90 by 90 block of numbers. This has been checked out on Excel worksheets and proves to be true.
I said earlier that I would further discuss the 90 patterns and I will.
But first, since I have not yet learned LaTex, I am having to copy bits and pieces from my many pages of research. And I cannot copy and paste any charts or Excel worksheets to Math is Fun.
So, I would be willing to email my texts and charts to serious investigators like Bobbym and Zetafunc and perhaps others.

Other concerns:
I think I will be able to understand syntax and procedure in using LaTex itself.
But I have some questions.
Where can I practice during my learning process?
Can I do this on a Word document?
How do I display output and where?

I have been reading from Getting Started with LaTex by David R. Wilkins
I read this quote “Having created the input file, one then has to run it through the LaTeX program and then print it out the resulting output file (known as a `DVI' file).”
How does one run it through the LaTeX program?
Hoping for some advice.
Ron Altic