In row 1, the values of a[t] represent na for some c (e.g. RSA 100).

a[t+n] = nb

This is true for all c.

For the value of c, at the same t but in cell (-f,1), the value at a[t] = a(n-1) and a[t+n-1] = b(n-1)

This is the key.

The value of a[t] at -f and e in the first row have the same factor.

The values in each cell that have b as a factor are DIFFERENT, not aligned. They are one element apart in the two cells. In the positive side of the grid, they are n elements apart. In the negative side of the grid, the elements one less elements apart.

One cell has n as a factor at those positions at positive e column, one cells has n-1 as a factor in the negative f column.

This asymmetry can be used to solve the problem.

(e,1) has n as a factor

(-f,1) has (n-1) as a factor

Row (n-1)

Row n

At cell (e,n) c is an an element.

At cell (-f,n-1) c is also always an element

There is a pattern of repeating cells on each row. 2n in row n. 2(n-1) in row (n-1).

This gives additional information.

Information constrains values.

Every piece of information constrains values.

The most valuable information greatly constrains values as c increases.

Basic picture.

For all c.

c = ab = dd+e = (d+n)(d+n)-(x+n)(x+n) = aa + 2ax + 2an

Grid (p,q) where p and q are signed integers

Elements in a cell are products with notation: e:n:d:x:a:b

The first two of the notation correspond to the coordinates in the grid.

Horizontal black line (e,1), (-f,1)

Vertical black line (0,n)

Vertical grey line (-1,n)

For some SPECIFIC c = ab = dd+e = (d+n)(d+n)-(x+n)(x+n) = aa + 2ax + 2an

Dark green line : column that contains e

Dark maroon line : column that contains -f

Pinkish-purple square cell in dark green line at (e,1) contains an and bn at elements t and t+n which are elements:

e:1:(na+x):x:na:(na+2x+2)

and

e:1:(nb+x+2n):(x+2n):nb:(nb+2x+4n+2) (editor's note: this is a correction made by PMA. It was originally e:1:(nb+2x+2n):(x+2n):(nb+x+2n):(nb+3x+6n+2), then VQC changed it to e:1:(nb+2x+n):(x+2n):(nb+x+2n):(nb+3x+6n+2), and then when PMA said it was the first thing VQC replied with "Thank you").

Blue square in dark maroon line (-f,1) that contains a(n-1) and b(n-1) at t and t+n-1 elements

Orange squares in -f line and e line : squares that contain c as a product… -f:n-1:d:x:a:b and e:n:d:x:a:b respectively. THESE SQUARES ARE ONE LINE APART.

Pick any odd c and this holds for all. ALL.

From (-f,n-1) = c, the value of a,b and d increase by ONE every 2(n-1) cells, as you move from left to right in the grid.

From (e,n) = c, the value of a,b and d decrease by ONE every 2n cells, as you move from right to left in the grid.

These two rows are next to each other.

Any product of 2 primes will be divisible by 3 if you add either 2 or 4, or if you subtract 2 or 4.

Any product of 2 primes will be divisible by 5 if you…

Etc…

Remainders.

Patterns.

"Triangulation"