Harristotelian Logic

To understand the end of the mathematical debate you need something simple:

If you have numbers f and g where

fg = 3

and you are in a ring where f can be coprime to g, and f is coprime to g, and you have another number h, where h is coprime to g, then h and f must share the same factors in common with 3.

Now your base expression is

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

and first note that

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

are coprime as you can just subtract one from the other to see, which forces

5 + (sqrt(37) - sqrt(13))/2 to share factors in common with 3 with sqrt(37) + sqrt(13) because

(sqrt(37) + sqrt(13))(sqrt(37) - 13)/4 = 6

and refer back to the simple concept I started with to fully understand.

Now simply note that

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

forcing 5 - sqrt(13) to have factors of 3 in common with sqrt(37) + sqrt(13) as remember

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

does as well as just proven.

Now finally use the base expression to get

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

and note that 5+sqrt(37) is now forced to have those same factors of 3, by repeated use of the simple result at the top of this post.

Now you also get the result that sqrt(37) - sqrt(13) cannot have factors in common with 3.

I want you to now go out on the newsgroups and read replies of posters in various threads I've created.

These people have controlled many of you simply by being disagreeable.

It is human nature to look to others to see what they are doing first before acting.

They just always reply in the negative and thereby control you.

It's quite easy, quite simple and notice, does not require them to be right.

They just disagree with me, and your inclination—your human need—is to follow what you perceive to be the group opinion, and they know this, and use your humanity against you.

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I discovered more problems with my earlier argument but also found the fix as I was thinking about it all while watching the World Series game tonight.

Starting still with

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

is the same as

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

proving that 5 + sqrt(37) must share the same factors from 3 with 5 + (sqrt(37) - sqrt(13))/2).

Now I use a second relationship

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

and now divide that by the previous

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

to get

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

where from before I have that (5 + (sqrt(37) - sqrt(13))/2) shares all its factors in common with 3 with ((sqrt(37) + sqrt(13))/2), so those cancel out.

That proves that (sqrt(37) - sqrt(13))/2) can only share factors in common with (5 + sqrt(37))/2 and must be coprime to (5 - sqrt(37))/2.

But, since ((sqrt(37) + sqrt(13))/2)((sqrt(37) - sqrt(13))/2)) = 6, it cannot be true that any factors in common with 3 are being divided out, so

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

is not allowed to actually have any factors in common with 3.

Kind of weird in a way as the correct answer is the opposite of what I thought it was before.

I just needed the second relationship to get it right.

But regardless, there it is, finally the coup de grace.