Science Forums 2

CraigD · 10-07-2008

Quote:

Originally Posted by Qfwfq

Put your mouse pointer over some of Craig's equations and you will see the code (when it isn't too long); you should be able to figure enough for your purposes here.

Even easier, rather than clicking on the [New Reply] button, Click on the [quote] button at the bottom of a post containing a rendered equation. You can then copy (in most browsers, highlight the [math]…LaTeX stuff…[/math], rightclick and chose Copy) the equation. LaTeX is pretty intuitive – though little if at any easier to read before it’s rendered than plain text with lots of ()s.

The quote button also provides a nice alternative to posting text like "In post #n, xxx said:"

Quote:

Originally Posted by Qfwfq

Quote:

Originally Posted by CraigD

This step is algebraically incorrect.

Er, I don't see the mistake in:

$\left(\frac{T}{T}a^\frac{x}2 \right)^2 +\left(\frac{T}{T}b^\frac{y}2 \right)^2 = \left(\frac{T}{T}c^\frac{z}2 \right)^2= T^2 \left(\frac{c}{T} \right)^{\frac{\frac{z}2\frac{\ln(c)}{\ln(T)}-1}{\frac{\ln(c)}{\ln(T)}-1}2}$

Quote:

Originally Posted by Don Blazys

This is algebraically correct. It is not a mistake.

Don and QfwfQ are correct – once again, I’m so disoriented by all the nested ()s in it, I’m not able to effective read Don’s math, and think it’s wrong when it’s not

I like the note in post #22, which elaborates prettily into
$a^x = T \left(\frac{a}{T}\right)^{\left( \frac{x\log_Ta -1}{\log_Ta -1}\right)} = T^n \left(\frac{a}{T}\right)^{\left( \frac{x\log_Ta -n}{\log_Ta -1}\right)}$

However, I don’t’ see how these identities provide or give any direction toward a proof or disproof of Fermat’s last ( $a^n +b^n \not= c^n, a,b,c,n \in \mathbb{Z}, a,b,c>0, n>2$ or Beal’s conjectures ( $a^x +b^y = c^z, a,b,c,n \in \mathbb{Z}, A,B,C>0, x,y,z>2$ only if $A, B$ and $C$ have a common prime factor).

These conjectures are challenging because they are restricted to the integers ( $\mathbb{Z}$ ), not the reals ( $\mathbb{R}$ ). If $a,b,c \in \mathbb{R}$ , it’s trivial to disprove either conjecture by constructed example (eg: $1^3 +1^3 = (\sqrt[3]{2})^3$ )

Like any logarithm function,
$a^x = T \left(\frac{a}{T}\right)^{\left( \frac{x\log_Ta -1}{\log_Ta -1}\right)}$
Is true of for $a,x,T \in \mathbb{R}$ , requiring only $T \not=a,1$ and $T,a>0$ . I don’t see how it can be restricted to the intergers in some way that makes it useful for proofs of Fermat’s last, Beals, or similar conjectures.

Don seems to me to be hinting that $\frac{a}{a} \not=1$ and/or $1 \cdot a \not= a$ , negating a couple of the most fundamental postulates of arithmetic, which makes no sense to me at all. Also, I can’t make sense of statements like

Quote:

Originally Posted by Don Blazys

Had mankind learned how to properly represent and eliminate common factors to begin with, (using at least one "cohesive term" to ensure that any common factor is "non-trivial") problems such as the BC and FLT would never have surfaced!

Fermat’s and Beale’s conjectures are either true or false (for the handful of people who actually understand the proof, Fermat’s last conjecture actually is true). Even if proven, how could these and similar conjectures not “surface”?

Either I’m missing something, or Don’s hints really are leading nowhere.

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