The modular inverse is a rather simple thing from modular arithmetic, like consider:
2(3) ≡ 1 mod 5
Here 2 is the modular inverse of 3 and vice versa because they multiply to have a residue of 1 modulo 5, which is the modulus. That can be written as:
2 ≡ 3-1 mod 5
Importantly in my research as I use SO much modular long ago tired of copying and pasting the modular congruence symbol so just use equals, so I have:
2(3) = 1 mod 5 and then: 2 = 3-1 mod 5
May seem small but in my experience you can face purists who will be dismissive on such small matters! When am someone who is NOT going to waste time copying and pasting something all over a vast amount of research just to appease such people as is human convention. The math does not care.
The modular inverse as a concept has been around for some time, but only a few basic approaches for finding it were previously known. One relies on something called the extended Euclidean algorithm, which is very simple but will just link. The other approach depends on something called Euler's theorem, where learned the above from article on Wikipedia on what they call the modular multiplicative inverse.
And now can add another basic approach which relies on a system of equations I discovered about a month ago. For some residue r modulo N, its modular inverse is:
r-1 = (n-1)(r + 2my0) - 2md mod N
Where y0 is chosen as is m, with m not equal to r, and n and d are to be determined. They are found from:
2mdF0 = [F0(n-1) - 1](r + 2my0) mod N
F0 = r(r+2my0) mod N
Copied from my post: Modular Inverse Innovation
It is also derived there. And my system, is kind of more direct like that which follows from Euler's theorem, but is also iterative like what follows from extended Euclidean algorithm. But allows you to fiddle with things in a way that neither does. So you can pick two key variables: m and y0
And they are so named because of how I discovered the system, as was just kind of puzzling over some things.
It is one of my most direct discoveries which surprised me a bit, as it just flowed. And I'm beginning to accept that I have years of experience which has given me a certain level of expertise.
Research about the modular inverse on the web I've done since, when yeah got REALLY interested, has indicated is also a practical result as calculations of the modular inverse are part of modern techniques, according to that research. So it has applied and pure math aspects.
The result is definitive though in terms of evaluating social aspects of how discovery is actually treated versus how one might imagine. I've had lots of experience in this area with prior results.
And the social problem label below covers that topic.
Good news though, looks like first basic major result at this level in over a century. Probably MUCH longer. Notice prior methods go back to Euler and Euclid. Wow. Supposedly such a possibility of such a basic find no longer existed. Am thrilled to be the discoverer.
So yeah, humanity has a new mathematical tool.
Mathematics discovered belongs to the human race.