With my math around what I decided to call tautological spaces, which are complex identities used modularly, have certain ways of doing things where did look to get enough needed terminology written down. So yes, there is an invention there. Figuring out terminology is fun to me. Is work though too.

The innovation is of extreme interest considering its demonstrated power. Gist of it is so easy though:

x+y+vz = 0(mod x+y+vz)

Which is an equivalence written as a modular expression and introduced world's first true modular algebra in contrast to modular arithmetic. Which is about all those variables in the modulus. Actually DID go looking around to see if could find ANY math out there similar.

I write that and ponder. These are things have considered for more than a decade but often thinking quietly to myself, though leaking out here and there in posts. But now am simply stating these things completely.

Is good to be able to consider after over 18 years. And before there were fears, like maybe would just not amount to much really. And think really was 2008 when felt like, oh, is solid. And of course could check as hey, have my very useful definition of mathematical proof. Which was motivated by NEED, where am thankful there. So knew was absolutely correct, as had checked already. Still as human beings we have emotions that can take time to settle down. So now is even better about a decade later. Am calm on much more.

There are a couple of innovations there. So there is using modular for an equivalence which is so cool, realizing is equivalent to: x+y+vz = x+y+vz

Also there is the asymmetric form. I tried symmetric forms first! Which does not surprise me. And naturally was starting with x,y and z, where point of innovation was to add an additional variable, but is intriguing that you NEED at least three variables which at one point had me contemplating if is related to why our...heading off into physics.

Could talk more about mathematical innovations have introduced where could get a VERY long post, but want a short one, so will talk just one more:

dS(x,p

_{j}) = [x/p

_{j}] - 1 - (j-1) - S(x/p

_{j}, p

_{j}-1)

where S(x/p

_{j}, p

_{j}-1) is the count of composites that multiply times p

_{j}to give a product less than or equal to x, where notice that p

_{j}must be less than or equal to sqrt(x) or the composite count given by [x/p

_{j}] - 1 will not be correct.

So the dS function is the count of composites for a particular prime excluding composites that are products of lesser primes.

Reference: Composite counting functions and prime counter

Where the innovation is explained with that last. And remember my thinking back summer of 2002 when was looking for something to count primes. And distinctly remember thinking it silly to count composites already counted by a lesser prime. Like with composites up to 10, when considering those with 3 as a factor, why bother with evens? Those are counted by 2, and I mathematized that viewpoint.

Have stared at such things through the years, and pondered what it takes. And in my case? Admittedly lots of training with modern problem solving techniques. Was trained to work to think in a way most understand with the phrase--thinking outside the box. And was trained in brainstorming techniques.

Where plenty of people are so trained--in our modern times. Guess does make the difference where that training is applied and to get lucky. For me a few key innovations helped me help our world understand vast areas of mathematics with more powerful, and simplifying concepts.

That just keeps getting better too.

James Harris

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