To what audience then?

Have wanted to move from posts which to me were trying to push certain people to acknowledge what am certain are important mathematical truths. But it can be hard not to be emotional, which part of me thinks is a path to being more convincing. However philosophically prefer to stick to the value of simply expositing the truth.

By the rules established my most compelling results for others should be around a published result. Where feel important to emphasize how wacky that got back in 2004. And recently emphasized was demonstrating how a correct argument under all the established rules of mathematical rigor could nonetheless lead to a wrong conclusion!

That result is as valid today as was back then, of course. And is simply a demonstration of how declaring the ring to be the ring of algebraic integers can lead to a result outside that ring despite starting with expressions valid in the ring and only using ring operations.

More recently defending that result found myself fascinated with a general factorization where have repeated over and over but also I like the math.

In the complex plane: 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) does not equal 0 for all x. The simplest example is: P(x) = x² + 3x + 2

Can solve with some simple things: g₁(x) = f₁(x)/k and g₂(x) = f₂(x) + k-2

Multiply both sides by k, and substitute for the g's--to force symmetry:

k*P(x) =  (f₁(x) + k)(f₂(x) + k)

Forcing symmetry lets me solve after introducing a handle function: f₁(x) + f₂(x) = H(x)

Then have a defining quadratic:

f₁²(x) - H(x)f₁(x) - kH(x) - k² + k*P(x) = 0

Can now solve using quadratic formula but main thing is that with integer k, have algebraic integer solutions because is a monic quadratic with integer coefficients then.

And with k = 1 or -1, you are forced to have algebraic integer solutions for the f's and the g's. Ok well that's cool.

But move to other integers like k = 2, and g₁(x) = f₁(x)/k blocks one of the g's from being an algebraic integer if P(x) is irreducible as then have: g₁(x) = f₁(x)/2

(Oh, if get confused can use simplest P(x) which IS reducible to get a handle on what is happening.)

That is so devastating to so much of prior thinking in mathematics.

But with such elementary methods, backing up a published proof.

So why does nothing happen now to officially address and fix the problem? How can a mathematical community continue in error against such simple proof?

Will leave as rhetorical. At least is a pure math result, so had no impact on applied mathematics. Which means doesn't matter for our science and technology at THIS point at least. There may be applied mathematics down the line, yet to be discovered though from the corrected mathematics.

The wild thing then is that we now are looking at one of the g's which is still integer-like but cannot be an algebraic integer, so what is it?

Years ago I pondered. And pondered, and came up with something. That something is talked about in the first post for this blog.

These new to us numbers have intriguing properties. Or I say new to us, as in new to humanity in general, while now have been contemplating them for...how long? Guess really since 2003 before my paper.

Were the reason really for that paper. New numbers previously not catalogued. And a human species still not quite fully coming to grips with them.

But the math does not care. To the extent these numbers rule? They do not care what we think.

Will still work at shifting the tone as ponder what audience finds the math to be of interest, and push myself to give up on others who do not.

You cannot force true curiosity and why bother?

Mathematics can demand the most the human mind can bring to barely understand. Without interest?

There is no way the math will make sense at the outer limits of human knowledge without working hard for that understanding.


James Harris