The virtual quantum computer (VQC) is a grid made of infinite yet constructable sets that follow a known pattern. Like a quantum spreadsheet.

The grid is the superposition. The collapse of that superposition will be two input parameters, d and e which can be calculated easily for all integers, c, where c is the difference of two squares. Its purpose and our goal is to learn and show the TRUTH, one of them being P=NP. Cracking RSA will be a consequence.

When the integers that are the difference of two squares are arranged into the grid and their corresponding properties are shown, a pattern emerges that shows calculation instead of searching is possible.

==Glossary==
‘’Column’’
All cells for a given e.

‘’Row’’
All cells for a given n

‘’’The grid is indexed using e, n, and t, where e is the rows, n is the columns, and t is the specific element in the cell.’’’

‘’Entry, record, element’’
one set of variables that represents one factorization for a number.
an entry = {e:n:d:x:a:b} (e, n, t)
{1:5:12:7:5:29} (1, 5, 4) is a record AKA an element AKA an entry.

‘’ab record, nontrivial factorization, prime record’’
the element that contains the factorization of c that is not 1*c, hence, nontrivial.

‘’1c record, trivial factorization’’
the element generated from setting a=1 and b=c

‘’Cell’’
All entries for a given e,n (not to be confused with an entry itself.)

‘’Genesis cell’’
e,1

‘’Remainder Tree’’
The remainder tree is the result of treating d and e as c's recursively until 1 is reached, creating a tree with several to many branches.

‘’’Functions’’’

‘’na transform’’
a movement from a record in (e, n) into (e,1) where n becomes 1 and a becomes a times the n of the (e,n) record. It has also been used to refer to moving n*a records down in a cell.

‘’T’’
T of number or T(input) is the triangle number function. If our input is 7, T(7) returns the 7th triangle number

‘’T-1, inverse T’’
the inverse function of the triangle number function that returns the index of a given triangle number. If our input is the 7th triangle number, the function returns 7.

Variables

The map's legend is {e:n:d:x:a:b}, where c is any number that is the difference of two squares, so odd numbers are included. It is the number you want to factor. It is the number that the a and b in an entry multiply to make.
a and b are, to reiterate, the factors of c. a is the smaller factor of c, and b is the larger one.
d is the integer square root of c
e is the remainder of taking the integer square root of c. Unless c is a perfect square, a remainder will be left over.
i is the root of the large square. it is the same thing as (d+n)
j is the root of the small square. it is the same thing as (x+n). i^2 - j^2, difference of squares.
n is what you add to d to be exactly halfway between a and b, and it is the root of the large square. So it takes you from d to the large square.
x is what you add to a to make d. When added to n it makes the root of the small square.
f is what you add to c to make a square. (e is what you subtract from c to make the square below it, f adds to make the square above c.)
g is the square root of c with decimals, opposed to d, which discards decimals.
t is the third coordinate in the VQC, it is a function of x.
u is the base of a triangle that helps us calculate (x+n) for certain c values. simply put, it is a representation of (x+n). 8 times the triangle number of u plus one is x+n.
s was a variable used to demonstrate patterns in (e, 1). See "(e, 1)."

When capitalized versions of the variables are used in comparison to lowercase versions, the capitalized versions refer to the variable's value for the trivial record, and the lowercase variables refer to the values for the nontrivial record.
{e:N:d:X:A:B} (e, N, T) is the trivial element.
{e:n:d:x:a:b} (e, n, t) in this context is the nontrivial element, the prime factorization of c.


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==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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==Code==

''C#''
'''BigInteger Square Root'''  ——  ''https://''pastebin.com/rz1SdACZ
'''VQC code w/ Bitmap'''  —— '' https://''pastebin.com/hMTtJF6E
'''PMA's tree generator'''  ——  ''https://''pastebin.com/ZH9fSWu2
'''Original VQC code'''  ——  ''https://''pastebin.com/XFtcAcrz
'''Unity Script'''  ——  ''https://''pastebin.com/QgAXLQj3
'''Unity Script 2'''  ——  ''https://''pastebin.com/Y38nVWgT

''Java''
'''Traverse the VQC cells in real-time'''  ——  ''https://''anonfile.com/TeH6q3d8bd/VQCGUI_v2.7z
'''Tree Generator'''  ——  ''https://''pastebin.com/VZnQQR2i
'''VQCGenerator'''  ——  ''https://''pastebin.com/Dgu9aP1h
'''VQC Triangle Number Methods'''  ——  ''https://''pastebin.com/NCQ3HK2K

''NodeJS''
'''BigInteger Library and Sqrt'''  ——  ''https://''pastebin.com/y8AXtFFr

''Python''
'''3D VQC [V2]'''  ——  https://pastebin.com/wZM5Thzu
'''Useful methods from CollegeAnon'''  ——  ''https://''pastebin.com/d8xZZnm0
'''Create the VQC'''  ——  ''https://''pastebin.com/NZkjtnZL
'''Fractal cryptography'''  ——  ''https://''pastebin.com/XuN4U7Dv
'''GAnon's Viewable Grid code'''  ——  ''https://''pastebin.com/czpK8A4j
'''Generate any cell in (0,1) and (0,2)'''  ——  ''https://''pastebin.com/gRTYpdMU
'''Generate cells for a (and more)'''  ——  ''https://''pastebin.com/iAizgLFF
'''Generate genesis cell'''  ——  ''https://''pastebin.com/GKzcCpMF
'''Generate positive AND negative genesis cells'''  ——   ''https://''pastebin.com/9ixjRyxt
'''VQC + t'''  ——  ''https://''pastebin.com/Lgufk0db
'''RSA & PGP key wrapper'''  ——  ''https://''pastebin.com/vNqnPRJR

''Rust''
'''Additional VQC code'''  ——  ''https://''play.rust-lang.org/?gist=50def916ad48400bc5d638fbf119ae85&version=stable
'''Check if a number is prime'''  ——  ''https://''huonw.github.io/primal/primal/fn.is_prime.html
'''Create Bitmap using the VQC Generator [V2]'''  ——  ''https://''pastebin.com/zGSusyz5
'''Generate the VQC'''  ——  ''https://''play.rust-lang.org/?gist=6b6beb372b6b931f1abd30642a35a80c&version=stable

'''Static Java/C# class with all RSA numbers'''  ——  ''https://''pastebin.com/XYFpsDWE

==Factorization methods (Java)==
'''Binary search for i'''  ——  ''https://''pastebin.com/TAt5bDsR
'''GCDFactor'''  ——  ''https://''pastebin.com/70GJSMrv
'''Calculate factors using -x jumps'''  ——  ''https://''pastebin.com/gKX9GW9r
'''Count down from t of 1c element'''  ——  ''https://''pastebin.com/xxYa946V
'''Mirrors 1c until e=(-x+n^2)'''  ——  ''https://''pastebin.com/WJBqPM4P
'''Shor's Algorithm (enter a random number < c as m)'''  ——  ''https://''pastebin.com/RD83RTNc

==Other Threads==
'''Fermat's Last Theorem'''  ——  ''https://''archive.fo/iTneU
'''Grid Patterns'''  ——  ''https://''archive.fo/isamV

'''RSA #0'''  ——  ''https://''archive.fo/XmD7P
'''RSA #1'''  ——  ''https://''archive.fo/RgVko
'''RSA #2'''  ——  ''https://''archive.fo/fyzAu
'''RSA #3'''  ——  ''https://''archive.fo/uEgOb
'''RSA #4'''  ——  ''https://''archive.fo/eihrQ
'''RSA #5'''  ——  ''https://''archive.fo/Lr9fP
'''RSA #6'''  ——  ''https://''archive.fo/ykKYN
'''RSA #7'''  ——  ''https://''archive.fo/v3aKD
'''RSA #8'''  ——  ''https://''archive.fo/geYFp
'''RSA #9'''  ——  ''https://''archive.fo/jog81
'''RSA #10'''  ——  ''https://''archive.fo/xYpoQ
'''RSA #11'''  ——  ''https://''archive.fo/ccZXU
'''RSA #12'''  ——  ''https://''archive.fo/VqFge
'''RSA #13''' —— ''https://''archive.is/Fblcs