## JSH: Understanding the arguments

Years ago I found a rather fascinating problem with the use of algebraic integers which is so upsetting that despite the ease of its proof, mainstream mathematicians have as far as I can tell almost completely ignored it, except for a brief publication of one of my early papers in the now defunct journal SWJPAM, where the editors yanked my paper after publication.

For those who wonder what the "crackpot" is arguing about with these people over and over again, year after year, I have this thread with a simple explanation using VERY elementary expressions and almost no complicated math, so that you too can understand the arguments.

Consider:

7(g_1(x) + 1)(g_2(x) + 2) = (f_1(x) + 7)(f_2(x) + 7) = 7*P(x)

where P(x) is a quadratic with integer coefficients, g_1(0) = g_2(0) = 0, and one of the f's equals 0 at x=0.

That construct is setup deliberately by me to indicate that the 7 only multiplies times g_1(x) + 1, so one of the f's has 7 as a factor, which is trivially true when you have polynomials. For instance, just change the functions to x:

7(x + 1)(x + 2) = (7x + 7)(x + 7) = 7*P(x)

and notice the 7 is FORCED to show up for one of the f's, if you are to stay in the ring of algebraic integers.

What I did though was figure out how to make the f's non-polynomial functions, with the same rules about 0, BUT if you try to put everything in the ring of algebraic integers, you find that there are situations where NEITHER of the f's can have 7 as a factor.

And THAT is the gist of it.

So how can people argue for years on such a thing? Well I say—there must be something wrong with the ring of algebraic integers then! Other posters say I'm wrong—or nastier things—and claim that there is nothing wrong with it.

And round and round we go!

Notice their idea is that somehow the FUNCTIONS can change how the 7 multiplies and often they would say that using x=0 is specious because they'd claim it was a "special case". And they'd go on and on about how the 7 could "split up" how it multiplied with:

7(g_1(x) + 1)(g_2(x) + 2) = (f_1(x) + 7)(f_2(x) + 7) = 7*P(x)

depending on the value of x.

I'd say the "tail does not wag the dog" and that the distributive property doesn't allow what's being multiplied to control how it is multiplied and that what is true for linear functions—where there is no argument—doesn't change because you shift to non-linear, non- polynomial functions, and they'd claim I was wrong!!! Or they'd claim it wasn't the distributive property. Or they'd claim anything rather than agree with me.

With a journal dead already, where I think it was over this issue, and with mainstream mathematicians not doing anything that I can notice, posters on Usenet produce most of the discussion of which I am aware, and there seems to be an established feeling that nothing will change soon!

Some of the posters who reply to me ask then why I bother to repeat the arguments that I say prove I'm right, if the group will never accept them, and of course, if I believe humanity will never accept the mathematical truth then human progress in mathematics ends.

Because the ideas here are trivially easy. There is no way the functions can push back outward to change how that 7 multiplies, so these errors are real, but well established. Worse, "pure math" arose—if you check you history—about the same time that the ring of algebraic integers arose, because I suggest, of the error!

So if this battle is lost, number theory progress for the human species, ends.

Which it may. So those who know the error is real, know that here and now they may be seeing the end of humanity's rise of knowledge in this area. The end of the growth that this species has had for thousands of years in mathematics.

The end of mathematical discovery happening right in front of them.

And how? Human nature is quirky. There is no rule that humanity must always progress necessarily as one day the human species itself will die. It will go extinct. So for those in the know, there may be an odd feeling at seeing one part of the human story end here and now. We at least can mourn its passing.

For those of us then, the discussion is worth having. And for us, hoping that humanity has a little more juice within it, is a hope worth having.

For the others though, the soul of mathematics is dead.