Harristotelian Logic

Wow that was hard. I had a feeling that the answer was in there somewhere but I kept getting these false positives only to figure out after a while that I was going down the wrong route, yet again.

Nearly thought it was over and there was no way this latest approach would work, but then I had the nice break watching the World Series and mulling things over, and FINALLY thought to use a second relation.

So I needed not only

(5 + (sqrt(37) - sqrt(13))/2)((sqrt(37) - sqrt(13))/2) = (5 - sqrt(13))(5 + sqrt(37))/2

but also

-(5 - (sqrt(37) + sqrt(13))/2)((sqrt(37) + sqrt(13))/2) = (5 - sqrt(13))(5 - sqrt(37))/2

so I could divide the second by the first and get the surprise finish.

And it took dividing too! As first I tried multiplying them together and didn't see a resolution.

Remarkably then the correct answer is that

5 + (sqrt(37) - sqrt(13))/2

can have NO factors in common with 3 as after you do that division and use the result that

5 + (sqrt(37) - sqrt(13))/2 must share any factors in common with 3 with sqrt(37) + sqrt(13), you find that it can't have any or sqrt(37) - sqrt(13) in the denominator is unbalanced.

Whew! Wow! What a surprise finish!

And so easy—once you know how to do it.

Oh yeah, so how did I get my neat relationships?

Check out my blog, as it's posted.

Question now is, how long will math people continue to defy mathematical truth?

Will any of the posters who worked so long and hard to deny my results finally concede to mathematical reality?

Will any of the top mathematicians hold a press conference quickly, or will they try to run?

I want this resolved by tomorrow. And I strongly suggest that mathematicians who wish to move forward to a good resolution make it their business to get a press release out there AS SOON AS POSSIBLE.

These delays have gone on for too long.

[A reply to someone who asked whether James was going to issue threats.]

I suspect they will.

The sci.math line has been broken, so they can't rely on you people any more.

You just look stupid at this point avoiding a result that is all about matching denominators.

People at the top in the field will now look to saving their skins and selling you out, I suspect.

And you will be easy targets and you are not anonymous, as governments can find out who you are, if you are using an alias or something other than your real name.

And I am saying that they need to find out everything about the posters who have been so dedicated—and effective—in helping to block my research, especially to see if any of you are actually top mathematicians who came here to actively block, not content to just hope the regular rank and file of sci.math were up to the task.

If any did, then they are likely definitely looking at jail time.

[James replies to his own first paragraph.]

And I was!!!

But I started thinking about why all those manipulations and decompositions were going wrong and finally had an idea, pursued it and here it is.

It is the coup de grace:

((2 + sqrt(5) + sqrt(13 + 4sqrt(5)))/2 + (-5 + sqrt(29))/2 + 5)((2 + sqrt(5) + sqrt(13 + 4sqrt(5)))/2 - (-5 + sqrt(29))/2) = (2 + sqrt(5) - (-5))(2 + sqrt(5) + sqrt(13 + 4sqrt(5)))/2

which is

((7 + sqrt(5) + sqrt(13 + 4sqrt(5))+ sqrt(29))/2)((7 + sqrt(5) + sqrt(13 + 4sqrt(5)) - sqrt(29))/2) = (7 + sqrt(5))(2 + sqrt(5) + sqrt(13 + 4sqrt(5)))/2

where, of course,

(2 + sqrt(5) + sqrt(13 + 4sqrt(5)))/2

is a unit, so only factors of 7 + sqrt(5) are available, where

(7 + sqrt(5))(7 - sqrt(5)) = 49 - 5 = 42

and those factors must divide through

sqrt(13 + 4sqrt(5))+ sqrt(29)

and

sqrt(13 + 4sqrt(5)) - sqrt(29)

on the right side, but multiply those two together and you get

13 + 4sqrt(5) - 29 = -16 + 4sqrt(5)

but -16 + 4sqrt(5) + 7 + sqrt(5) = -9 + 5sqrt(5), which is coprime to
3, showing that

7 + sqrt(5)

must be coprime to 3.

And with that—the math wars are over.