post 2 - rules

==Rules==
Each cell of the grid (e,n) has infinite elements or ZERO elements.
Each cell with one value has infinite elements, since every element can make a new one.
By induction, a cell only needs one value to make infinite values, that's part of the power of this and is why it is a virtual quantum computer as a whole.
The t variable is what will allow you to walk across these infinite elements.
If a grid cell has elements, all elements are constructable from a finite set of root elements.
Thus, only three variables are required to identify an element: e, n and t.
All products of odd numbers and all products of pairs of even numbers are the difference of two squares.

'''(1, 1) - the key'''
The values of a and b at 1,1 are related to the length of the longest side in right angled triangles.
The values here can be used to create the entire grid.
The values here determine the values of the rows to the left and right, which determine the values of the whole column.

'''Columns'''
Each cell at n=1 contains the roots of products in the column.
If c is a prime number, it will appear in one column exactly once.
If c is the product of two prime numbers that do not equal eachother, c will appear in two cells of one column.
All products (integers) c that are the sum of two squares appear (only) in columns where e=0,1,4,9,16,25...
All factors in a column are factors of the elements of the first cell in their column.
All Fermat primes (except) 3 appear in column one.

'''(e, 1)'''
If a number at position t has a factor s, then s is a factor at (t+s), (t+2s) and so on for a at (e,1).
Also, if a number at position t has a factor s at (e+1), then s is a factor at (s+1-t), (2s+1-t), etc for a at (e,1).
n*a and n*b for any c can be found n places apart in the cell at (e,1).

'''(1, n)'''
The cells in row one where n=1 have a relationship with the cells 2n to the right and 2n to the left.
Each "a" from the first row equals na because xx+e = 2na and na is half of that. That's BIG part of the KEY
Each element in a cell can be generated by moving up (t-1 = x-2) or down (t+1 = x+2). Other variables can be generated from x.

For more of these rules, see the grid patterns thread.

==Useful Equations and Notation==
ab = c
dd + e = c
(d + n)(d + n)-(x + n)(x + n) = c
a + 2x + 2n = b
a = d - x
d = a + x
d = floor_sqrt(c)
e = c - (dd)
b = c / a
n = ((a + b) / 2) - d
d + n = i
x = d - a
x = (floor_sqrt(( (d+n)*(d+n) - c))) - n
x + n = j
j^2 = 8*T(u) + 1
f = e - 2d + 1
u = (x+n) / 2

if (e is even) t = (x + 2) / 2
if (e is odd) t = (x + 1) / 2