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 Y2-X2. Redundant information can be removed by not showing the negatives. Since Y2-X2 = (Y-X)(Y+X), the only way to get a prime is when (Y-X)=1.

As a result all primes, except 2, will show up only in the (Y-X)=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 (Y-X)=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.

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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)

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