Belief can have an impact on discovery where have a great example I think from my recent realization about the three variable general reduced form for the quadratic case.
So the general equation is:
c1x2 + c2xy + c3y2 = c4z2 + c5zx + c6zy
where the c's are constants.
And noted recently could show where was just as easy to find Diophantine solutions for the three variable as for the two variable case! Is kind of wild really. Where have had the primary argument since 2008 so recently just recognized it. Where I guess that's new. The general reduced form for the three variable case is just: u2 - Dv2 = Fw2
Where yeah, humanity has had the ability in principle to so reduce since November 2008, since I'm a human and I'd figured out how, but then wandered off to other things before fully realized.
The simple innovation to reducing a seeming VASTLY complex expression? I subtract it from a complex identity I call a tautological space, and that space does complex algebraic manipulations--for me.
Of course interested folks can go out and check if there is anything else out there to generally reduce three variable quadratic equations. Tried briefly myself and found nothing which does not mean is not out there. But if not, consider just how much it can matter to ignore discovery, especially if you are someone who needs, yup, to generally reduce a three variable quadratic! That is a pure and applied mathematical discovery.
Another favorite of mine where innovation simply surprised is around 16 years old, as thought about counting primes. And yup, noted as did others centuries ago that with a prime p, you can count composites by dividing by it and subtracting one. Like up to 10 there are 5 evens: 10, 8, 6, 4, and 2 itself. And up to 10 there are 3 numbers with 3 as a factor: 9, 6 and 3
But was like, why bother counting that even 6 again? And figured out the math will let you automatically count at each prime excluding counts from all lesser primes, which gives an innovative form for the composite count.
Going to copy much from a prior post, and add some highlighting with bold, but main thing is, notice the simple mathematics! But that math does what I just noted above--counts composites at each prime without bothering with counts from smaller primes.
(Should note below that [] is the floor() function. For example [10/3] = 3. So is just dividing without bothering with the remainder if any.)
The main workhorse is the dS function though which is where I had to do the most work in figuring out its form:
dS(x,pj) = [x/pj] - 1 - (j-1) - S(x/pj, pj-1)
where S(x/pj, pj-1) is the count of composites that multiply times pj to give a product less than or equal to x, where notice that pj must be less than or equal to sqrt(x) or the composite count given by [x/pj] - 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. So it gets a count of integers with a prime as a factor, subtracts 1 for the prime itself, and then subtracts the count of primes less than it. And finally it subtracts the count of composites multiplied times that prime.
Reference post: Composite counting functions and prime counter
Where not only do I get a simpler mathematics for counting primes, weirdly enough, but it also leads to calculus and an explanation of why you connect to continuous functions.
Mathematics is logical. We humans can BELIEVE something is hard, but does not make it hard because to the math? Nothing is hard.
To the math all mathematics just is. There is no such thing as hard or easy from the perspective of, the math.
One advantage I think I have is a healthy disrespect for our abilities as human beings.
We cannot compete or compare with the math. We can only learn from the math.
And for us humans belief can matter much! Like I wandered off from a three variable reduction for years thinking wrongly that x, y and z were harder than just x and y. But had so much fun like with my BQD Iterator that I don't regret it.
While with my prime counting function lucked out that I didn't even bother with what was known before struggling at figuring out my own way, and found that my common sense and the math agreed!
The math DID have a way to do something that made sense! And it dawns on me that maybe there are humans who do not respect infinite intelligence.
We can figure out some things, but the math knows ALL. If something makes sense to us? If it actually does make more sense then there may very well be a mathematical way.
But you have check to know. The math does not care what we know.
Glad I checked. And humanity gets more tools as a result. Is win-win.
So why does one human out of BILLIONS find such things? Who knows really but I think is because I believed simple solutions might be there, so I went looking for them. Is interesting how much belief can impact our lives and determine what we know and even what we can discover.
Better thing I think with math is--try something. Especially as can do so quietly, if only just curious.
And who knows? Maybe you will find something regardless worth sharing. But even the exercise can be good for the mind.
James Harris
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