So I promised a post after work last night, discussing the new factorization technique.

As before, I use a method called decon() that takes any number, like 697 for example, and first breaks it down into the respective digits and magnitudes.

697 becomes -> 600, 90, and 7.

It then factors *those* to give a decomposition matrix that looks something like the following when printed out:

offset: 3, exp: [[Decimal('2'), Decimal('3')], [Decimal('3'), Decimal('1')], [Decimal('5'), Decimal('2')]]

offset: 2, exp: [[Decimal('2'), Decimal('1')], [Decimal('3'), Decimal('2')], [Decimal('5'), Decimal('1')]]

offset: 1, exp: [[Decimal('7'), Decimal('1')]]

Each entry is a pair of numbers representing a prime base and an exponent.

Now the idea was that, in theory, at each magnitude of a product, we could actually search through the *range* of the product of these exponents.

So for offset three (600) here, we're looking at

2^3 * 3 ^ 1 * 5 ^ 2.

But actually we're searching

2^3 * 3 ^ 1 * 5 ^ 2.
2^3 * 3 ^ 1 * 5 ^ 1
2^3 * 3 ^ 1 * 5 ^ 0
2^3 * 3 ^ 0 * 5 ^ 2.
2^3 * 3 ^ 1 * 5 ^ 1

etc..

On the basis that whatever it generates may be the digits of another magnitude in one of our target product's factors.

And the first optimization or filter we can apply is to notice that assuming our factors pq=n,
and where p <= q, it will always be more efficient to search for the digits of p (because its under n^0.5 or the square root), than the larger factor q.

So by implication we can filter out any product of this exponent search that is greater than the square root of n.

Writing this code was a bit of a headache because I had to deal with potentially very large lists of bases and exponents, so I couldn't just use loops within loops.

Instead I resorted to writing a three state state machine that 'counted down' across these exponents, and it just works.

And now, in practice this doesn't immediately give us anything useful. And I had hoped this would at least give us *upperbounds* to start our search from, for any particular digit of a product's factors at a given magnitude. So the 12 digit (or pick a magnitude out of a hat) of an example product might give us an upperbound on the 2's exponent for that same digit in our lowest factor q of n.

It didn't work out that way. Sometimes there would be 'inversions', where the exponent of a factor on a magnitude of n, would be *lower* than the exponent of that factor on the same digit of q.

But when I started tearing into examples and generating test data I started to see certain patterns emerge, and immediately I found a way to not just pin down these inversions, but get *tight* bounds on the 2's exponents in the corresponding digit for our product's factor itself. It was like the complications I initially saw actually became a means to *tighten* the bounds.

For example, for one particular semiprime n=pq, this was some of the data:

n - offset: 6, exp: [[Decimal('2'), Decimal('5')], [Decimal('5'), Decimal('5')]]

q - offset: 6, exp: [[Decimal('2'), Decimal('6')], [Decimal('3'), Decimal('1')], [Decimal('5'), Decimal('5')]]

It's almost like the base 3 exponent in [n:7] gives away the presence of 3^1 in [q:6], even
though theres no subsequent presence of 3^n in [n:6] itself.

And I found this rule held each time I tested it.

Other rules, not so much, and other rules still would fail in the presence of yet other rules, almost like a giant switchboard.

I immediately realized the implications: rules had precedence, acted predictable when in isolated instances, and changed in specific instances in combination with other rules.

This was ripe for a decision tree generated through random search.

Another product n=pq, with mroe data
q(4)
offset: 4, exp: [[Decimal('2'), Decimal('4')], [Decimal('5'), Decimal('3')]]

n(4)
offset: 4, exp: [[Decimal('2'), Decimal('3')], [Decimal('3'), Decimal('2')], [Decimal('5'), Decimal('3')]]

Suggesting that a nontrivial base 3 exponent (**2 rather than **1) suggests the exponent on the 2 in the relevant
digit of [n], is one less than the same base 2 digital exponent at the same digit on [q]

And so it was clear from the get go that this approach held promise.

From there I discovered a bunch more rules and made some observations.
The bulk of the patterns, regardless of how large the product grows, should be present in the smaller bases (some bound of primes, say the first dozen), because the bulk of exponents for the factorization of any magnitude of a number, overwhelming lean heavily in the lower prime bases.

It was if the entire vulnerability was hiding in plain sight for four+ years, and we'd been approaching factorization all wrong from the beginning, by trying to factor a number, and all its digits at all its magnitudes, all at once, when like addition or multiplication, factorization could be done piecemeal if we knew the patterns to look for.