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# post 2 - rules

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Sep 29th, 2018
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__Sign Up__- ==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

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