Harristotelian Logic

I'm going to go into detail explaining a simple bit of algebra that refutes quite a few ideas long-held in number theory:

(sqrt(13) - sqrt(6) + sqrt(41) + sqrt(34))(sqrt(13) - sqrt(6) + sqrt(41) - sqrt(34)) = 2(sqrt(13) - sqrt(6))(sqrt(13) + sqrt(41))

That can look complicated because of all the radicals, but you just need to take it in pieces:

(sqrt(13) - sqrt(6) + sqrt(41) + sqrt(34))

is your first factor on the left, while

(sqrt(13) - sqrt(6) + sqrt(41) - sqrt(34))

is the second, and you have on the right

2(sqrt(13) - sqrt(6))(sqrt(13) + sqrt(41))

where (sqrt(13) - sqrt(6)) and (sqrt(13) + sqrt(41)) are key as you can find each of them in the factors on the left.

So with the first factor on the left you can see sqrt(13) - sqrt(6) easily enough, but re-group within the same expression:

(sqrt(13) - sqrt(6) + sqrt(41) + sqrt(34))

becomes

(sqrt(13) + sqrt(41) - sqrt(6) + sqrt(34))

and now you see the second number sqrt(13) + sqrt(41) is also in there.

Nifty.

Now then, given that the right hand side has both numbers can each of the factors on the left NOT have factors in common with them?

I assume you say no, like how if you have xy = 6 where x and y are integers, can x and y be coprime to 6? No.

But then looking at the first factor the first way:

(sqrt(13) - sqrt(6) + sqrt(41) + sqrt(34))

does sqrt(13) - sqrt(6) necessarily share all its factors with sqrt(41) + sqrt(34)?

No.

But any factors NOT shared are then blocked, as let

f_1*f_2 = sqrt(13) - sqrt(6) and f_2*f_3 = sqrt(41) + sqrt(34)

where f_1 and f_3 are coprime, and then the expression is simply enough

(sqrt(13) - sqrt(6) + sqrt(41) + sqrt(34)) = f_2*(f_1 + f_3)

But you can do the same with the re-grouping:

(sqrt(13) + sqrt(41) - sqrt(6) + sqrt(34))

where this time I'll let

g_1*g_2 = sqrt(13) + sqrt(41), and g_2*g_3 = -sqrt(6) + sqrt(34)

so

(sqrt(13) + sqrt(41) - sqrt(6) + sqrt(34)) = g_2*(g_1 + g_3)

Now (sqrt(13) + sqrt(6))*(sqrt(13) - sqrt(6)) = 7,

while

(sqrt(13) + sqrt(41)*(sqrt(13) - sqrt(41) = -28

so both have factors in common with 7, and those factors must be shared between g_2 and f_2.

If you get that then the rest is trivial, as now you know that factors of 7 are shared between them in the first factor on the left, and the same argument can now be used with the second factor on the left:

(sqrt(13) - sqrt(6) + sqrt(41) - sqrt(34))

and since the two factors together multiply to give you both numbers:

(sqrt(13) - sqrt(6) + sqrt(41) + sqrt(34))(sqrt(13) - sqrt(6) + sqrt(41) - sqrt(34)) = 2(sqrt(13) - sqrt(6))(sqrt(13) + sqrt(41))

you know that they have the same factors in common with 7.

BUT now you can just use

z = (sqrt(13) + sqrt(41))/(sqrt(13) - sqrt(6))

to see if one divides into the other and find that it is a root of

7z^4 - 52z^3 - 100z^2 + 208z + 112 = 0

which is a non-monic polynomial irreducible over Q.

Which proves sqrt(13) - sqrt(6) is NOT a factor of sqrt(13) + sqrt(41) in the ring of algebraic integers!!!

BUT the simple argument above PROVES that sqrt(13) - sqrt(6) MUST BE a factor of sqrt(13) + sqrt(41), as they share the same factor in common with 7.

So there is an apparent contradiction, which is removed by simply noting that whether or not numbers are factors in the ring of algebraic integers is irrelevant, as guess what?

All it tells you is whether or not the result of dividing one by the other can give you the root of a monic polynomial with integer coefficients.

Now I've proven a problem with the ring of algebraic integers before, and even had a paper published, but some sci.math'ers mounted an email campaign, convinced the editors to yank the paper, and a few months later the journal quietly shut-down.

I re-wrote the paper somewhat and sent it to the Annals of Mathematics at Princeton. I was told it was accepted for review, but when I checked back six months later, I was told a rejection email had been sent out months before—but I never got an email, and still haven't gotten a rejection email to this day.

So why the big deal? Why would a journal collapse and die or Princeton break its own rules for handling papers over this issue?

Because it takes away ideal theory among some other very cherished ideas in number theory, and some people must have decided they'd rather teach students false mathematical ideas and hold on to the world as it is, than teach the truth.