Why new numbers, really?

Feel is important to continue to emphasize what is actually I guess my most dramatic discovery, which is easy to prove with elementary methods. So yeah back in 2003, realized that there were more important numbers than had previously been catalogued.

Eventually, thankfully, came across a very simple way to show their existence, with quadratics.

In the complex plane, consider:

P(x) = (g₁(x) + 1)(g₂(x) + 2)

where P(x) is a primitive quadratic with integer coefficients, g₁(0) = g₂(0) = 0, but g₁(x) is not equal to 0 for ALL x.

The simplest example is: P(x) = x² + 3x + 2

And that is the end of things just casually making sense. As by the time I knew to use this expression around 2010, also knew I NEEDED to multiply it by an additional, added factor I call k.

So, introduce k, where k is a nonzero, and new functions f₁(x), and f₂(x), where:

g₁(x) = f₁(x)/k and g₂(x) = f₂(x) + k-2

Multiply both sides by k, and substitute for the g's, which gives me the now symmetrical form:

k*P(x) =  (f₁(x) + k)(f₂(x) + k)

Getting to symmetry lets me solve for the f's, and do so as roots of a monic polynomial with integer coefficients, so I can make them be algebraic integers!

So yeah it is all a way to flip things cleverly. Is like building backwards. Then I get the crucial result which follows.

But for one of the f's: f₁(x) = kg₁(x)

Where now is built front and notice result NOW will apply in a ring.

And we have a problem if the f's are not rational. We do but the math doesn't. The algebra is straightforward. Is weird how easy it is really.

Of course indices are arbitrary so we simply know that one of the solutions has k as a factor. If you wonder why that is a problem, consider 3+sqrt(-26), as it can be shown that one of its two solutions has 7 as a factor!

Confused? Consider 1+sqrt(4) = 3 or -1. Tend to gloss over that quickly but it is bizarre that people might be taught any other way. It is actually simply mathematically incorrect to say that sqrt(4) is JUST 2, as -2 is a solution. You can't just wish away.

Oh yeah, so of course then 1 - sqrt(4) = -1 or 3. So has 3 as a factor for one of its two values.

But it can be proven that 3+sqrt(-26) also does NOT have 7 as a factor in the ring of algebraic integers.

Well yeah one of the g's cannot be an algebraic integer if k is some nonzero integer other than 1 or -1, when they are NOT rational. (Let them be integers like with simple example noted to notice how easy the math actually is.)

So yeah, my trick is forcing the f's to be algebraic integers, when I multiplied times some prior numbers, and that blows up LOTS of prior number theory when not rational. And then reveals new numbers! Yay!

The situation is like, if people knew of evens and did NOT know of odds, so you caught someone exasperated by the notion that 6 and 2 share 2 as a factor, because they don't know 3 exists.

Well there isn't a unit there which is a big difference. But situation is exactly the same with ring of algebraic integers except there are units.

But if you don't know there are those other numbers, yeah can get apparent contradictions which I exploited in my pivotal paper like to talk so much.

Like with this post: Publication does matter

Are these numbers a big deal? Of course. We're talking about number theory where MOST of the numbers, like an infinity of them, were previously unknown to exist! And I found them. Is cool.

(Oh my God is just so freaking amazing. Why me? How me? Oh my God, oh my God, oh my God.)

Am much better at handling that information now than in the past. See? I'm maturing as a human being.


James Harris