### Friday, December 04, 2009

I've found a rather potent error in some number theory, where it's kind of like a place where given 7x = 7x, the error stops you from dividing off the 7. But rather than explain it in detail I want you to ask a mathematician you know to do something that may seem simple. Which when you first see it may look trivial:

The request is, ask a mathematician to divide off the 7:

7(175x^2 - 15x + 2) = (5a_1(x) + 7)(5a_2(x)+ 7)

where the a's are roots of

a^2 - (7x-1)a + (49x^2 - 14x) = 0 in the ring of algebraic integers.

Now you do not need to know what an algebraic integer is, or why this particular request is so important, but you just need to be able to read the mathematician to see how he, or she reacts to this request.

The request shows a very devastating error that entered into number theory in the late 1800's but lurks in "pure math" areas so it probably doesn't effect much math that physicists use, though it may blow apart string theory, as people there go to more mathematical approaches and may have gone far enough to have this error impact, but you can give it to a string theorist as well as a mathematician and see what he does.

My fear is that your colleague, if the person is a colleague, will lie to you. He may be lying to himself.

But I think you will be able to tell.

That's why this test is so potent. It's easy to give. That expression IS kind of funky, with that 7 on the left hand side, so why can't it just divide off? Like with 7x = 7x, you can divide off, to get x = x. You can do it in complex number of course, but here it's a number theory issue where that "algebraic integers" part is important.

The expression IS an identity as it's a factorization.

So it is like 7x = 7x, fundamentally.

I think some of you know about this error because of my prior posts and you can see I'm taking a different tack. I think some of you deep down wish to see how a math person will react to this thing. Or maybe you know it IS a massive problem, so you will not do it, to try and protect fellow academics. Or out of your own fear of the response.

It's a fascinating error. How it has lurked in mathematics for over a hundred years is fascinating as well, as is the difficulty in getting mathematicians to face it. But what you do or do not do is important here.

It is not an academic exercise.