In the complex plane, here's the wacky construction:

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

And now normalize the functions, that is, have functions that equal 0, when x=0.

Looking at x=0, gives

a

^{2}+ a = 0, so a

_{1}(0) = 0, or -1, and a

_{2}(0) = -1, or 0.

So I can let a

_{1}(0) = 0, and then introduce a new function b

_{2}(x), where:

b

_{2}(x) = a

_{2}(x) + 1, so a

_{2}(x) = b

_{2}(x) - 1, and making that substitution:

7(175x

^{2}- 15x + 2) = (5a

_{1}(x) + 7)(5b

_{2}(x)+ 2)

where now a

_{1}(0) = b

_{2}(0) = 0.

So now I have easily that the 7 multiplied in a trivial way--which makes sense as I made up this example, why would I make it too hard?

So now it looks like we need one more functional substitution which is:

a

_{1}(x) = 7b

_{1}(x)

Which gives me:

7(175x

^{2}- 15x + 2) = (5(7)b

_{1}(x) + 7)(5b

_{2}(x)+ 2)

And I can divide off the trivial factor of 7 to find:

175x

^{2}- 15x + 2 = (5b

_{1}(x) + 1)(5b

_{2}(x)+ 2)

And in the complex plane, we're done!

Easy.

But um, we have this result now that one of the a's results by multiplying one of the b's by 7, and the a's are roots of:

a

^{2}- (7x-1)a + (49x

^{2}- 14x) = 0

But now if we want to play with our result, say with x=1, we find that the a's are: 3+sqrt(-26) and 3-sqrt(-26), and in fact if you try:

7(175(1)

^{2}- 15(1) + 2) = (5(3+sqrt(-26)) + 7)(5(3-sqrt(-26)) + 7)

You'll see it is correct! That math worked!

But it implies that one of the a's has 7 as a factor. But established number theory says

*neither*of the a's have 7 as a factor?

So what did I do wrong?

The construction is built using special techniques designed to make it valid in what I call the ring of objects which I figured out to handle this problem.

In the ring of objects then--as I figured it out BECAUSE of this problem--there is no problem at all. It says that 7 is indeed a factor.

It's kind of like 10 + sqrt(9) has 7 as a factor for one of its two solutions.

If you think 10 + sqrt(9) is 13, you're wrong, as it's 13 or 7 because sqrt(9) = 3 or -3.

James Harris