Prime Patterns in Simple Number Arrays

Original Post 2004 
 Can simple number arrays be used in the search for primes?
 Can other significant information be found there

Reformatted January 2010

Difference of squares charts
A difference of squares chart is easy
to define. Each cell is Y^{2}X^{2}. Redundant
information can be removed by not showing the negatives. Since
Y^{2}X^{2} = (YX)(Y+X), the only way to get
a prime is when (YX)=1. 

As a result all primes, except 2, will show up only in the
(YX)=1 diagonal. All odd composites will show up in other locations
also. This creates a new definition for odd primes: any number
that appears only once in the difference of squares chart 
and only in the (YX)=1 diagonal.
In the chart to the right, primes are marked with outlines.
Some other patterns are marked with color.
In this form the chart is an inefficient means of searching
for primes and other patterns because each composite occurs
more than once.


Related pages at this site:

Squares Charts
blue = perfect squares red
= a sequence of primes

Here again, if we start with the sequence of squares and add,
some patterns with primes occur. If the first row is extended
out to 43, you can mark one of the longest regular sequences
of primes that is known.
Again this pattern is plagued with redundancies.


We can eliminate the redundancies problem by drawing out squares
of increasing size and counting out the blocks.
Patterns appear to be developing from this arrangement. However,
the sharp angles and the jump around the square (e.g.: from
25 to 26) lack elegance.


Outside Links

Top down view of number spiral cone. 
Can we build on these ideas by drawing our number line as a
spiral expanding around the surface of a cone? Appropriate definition
of our spiral will align the squares.
This will create a unique location for each number. Will the
spatial description of numbers prove beneficial?
What other sequences will this generate?



Goldbach Folding at Multiples
of 30 
We can draw from Goldbach's
Conjecture and fold our number line back and forth at multiples
of 30. 
primes are red,
multiples of 7 blue, multiples
of 11 green, 13 gray 

This groups our primes into 6 columns. However,
multiples of primes larger than 5 exist within our prime columns.
Can we find useful information in this pattern anyway? (only
the odds are shown) 



