When dealing with the issue of factorising a 2 co-prime composites.

So far I can find an upper and and lower bound for where I will find my lower co prime.
With that information I can then find a few (about 80 or so for RSA230) slots between where each of the few slots have a far smaller lower and upper bound and my co -prime will be somewhere between the upper and lower bounds of one of the slots.

This Math I am developing would greatly improve the amount of checking a general number filed sieve method requires.

I donno What the NSA/Chinese Intelligence has right now considering P almost definitely equals NP but for cracking 2 co-Prime Composites greater than 100 bits and smaller than 512 bits on the cloud and for some low bit numbers at home this work could become quite useful even for far faster factorisation below 100bits.

the 2 maths techniques I am using run at approx O 2(log N) and the other at approx O log(2,N)/log(3,N)
So they are fast but for very large numbers you would still be left with quite a range to check for your CO- Primes roughly N / Log(N) / (Log(N)^3) slots which is still a lot of number space to check for big Integers.

Alongside this I have already developed an approx O(13) run time between weak and strong Probable prime test which may or may not be of some use here.

I achieved this math with Wieta Temarkon and will be posting more on GitHub anonymously probably sometime next year.

I am but one maybe clever maybe mad person the NSA ect has most of the best mathematicians in such fields as cryptography. If you look on the Wikipedia at the RSA numbers section it Implies that a quantum computer could be built to tackle such things as maybe even Integers 4096 bits in size and if Edward Snowden has demonstrated with information they are watching everyone and everything they can and almost certainly are ahead of me when it comes to cryptography and aware already of all my work.

If you then look around the internet at people claiming RSA is fully hack-able they are laughed off on the given forums yet if you now look at the p vs np pages on the wikipedia or else where it is very hard to ascertain what exactly the fastest solution to an NP complete currently is as what ever it is it will be the fattest that any NP problem can be solved because NP complete problems can be used with very quick conversion to solve any NP problem.

To understand what an NP problem is lets look at chess. Now chess is not actually NP as far as anyone knows but chess is a good way to explain the issue. It's very easy to check if you have won your game of chess but very hard as far as we know to know the best move to make in any situation on the chess board. chess as far as is known is actually a exp complete problem and not NP but thats still up for debate and research as it is not known if chess is or isn't NP. an NP problem would be something like Sudoku now Sudoku is easy to check if you have won much like it is easy to check if a given Prime number divides into a given composite but harder to find a number that does fit.

Lets go a little more into depth.

so P time mean this say I had a P time algorithm that solved sudoku. my algorithm would look like this.
lets say we are dealing with a 81 square grid so x will be 81

Solver(x)
"Start"
If you have done function 1 x amount of times then stop and give result other wise do function 1

Funtion1(x)
do this x amount of times
then go back to start


So over all you will have done this x^2 times as you have x amount of calls to function 1 and function one does this x amount of times.

such a program would be O x^2 in sudoku's case this would be a maximum of 81^2 operations

Until recently at least no such program has existed for any of the NP complete problems like Sudoku
The only solutions were just below a maximum of O 2^x operations although for many 81 square sudokus a solution could be found far faster the maximum time it took was still in exponential time and thus thus some sudokus just couldn't be solved especially as the grid size was increased.

The whole point of all of this is that the whole internet was secured by the fact that such problems took to long in general to solve even with large computer farms so your information was to hard to break into.

However even if now this is not the case the whole issue of find better harder security is itself and NP problem.

So if NP has been cracked you might get senarios like joint cyber packs between previously hostile nations and a huge increase in military spending as to better secure military assets (which has happened already) this among other things like the TV picture seeming to look less choppy and OS systems seeming to operate far more optimally and such along with my own research leads me to believe that P = NP and that old quips about people who suspect such to be true are no longer fools especially as time goes by and I suspect that once the military asset's have been properly secured that there will be an announcement claiming that a second millennium problem has been solved probably by about 2023.