Virtual Quantum Computer
The virtual quantum computer (VQC) is a grid made of constructably infinite elements that follow a known pattern.
The grid in its entirety serves as the superposition. The input parameters that collapse the superposition are d and e, which are trivial to calculate for all c that is the difference of two squares. 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 a path to calculatethe factors of c instead of searching for them.
The grid is indexed using e, n, and t, where e is the column, n is the row, and t is the specific element in the cell.
Glossary
Look-up
A pattern used to calculate the factors of c, like a value look-up table.
Column
All cells for a given e
Row
All cells for a given n
Entry; record; element
A set of variables corresponding to a factorization for a given c. The legend to read entries is {e:n:d:x:a:b} (e, n, t) = c
Example: {1:5:12:7:5:29} (1, 5, 4) = 145
ab record; nontrivial factorization; prime element
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
Mirror element
The element in -f corresponding to an element in e, in the context of a given c.
Cell
All entries for a given e,n (not to be confused with an entry itself.)
Sieve
A sieve is an algorithm for factoring integers arising out of number theory in the 1900s, most notably from Carl Pomerance.
Smoothness
A number is described as smooth if it is composed of small prime factors, opposed to large ones.
Remainder Tree
The remainder tree is a structure created by recursively taking d and e starting with c, 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.
Pell(n)
The nth Pell number function. Can be calculated recursively or formulaically using the square root of 2.
ST(n)
The nth square triangular number. A square triangular number is a number that is both a square and a triangle number. The Pell function is used to calculate square triangular numbers.
T(n)
The triangle number function.
Example: T(7) = the 7th triangle number
T-1(n), inverse T
The inverse triangle number function.
Example: T(7th triangle number) = 7
Variables
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 equal to (d+n).
j is the root of the small square. it is equal to (x+n).
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.)
t is the third coordinate in the VQC, it is a function of x.
q is a product created by multiplying successive primes until the product is above d.
u is the triangle base of (x+n)^2. 8 times the triangle number of u plus one is (x+n)^2 for c with odd x+n.
h was a variable used to quantify families of numbers. The way to calculate it is currently unknown.
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. Sometimes these trivial uppercase variables are referred to with "Big" preceding the letter.
{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.