My Math Forum Cohesive Term Theory

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 April 4th, 2009, 08:49 PM #1 Newbie   Joined: Mar 2009 Posts: 12 Thanks: 0 Cohesive Term Theory "Cohesive Terms" or "Blazys Terms" as they are otherwise known, were invented by yours truly about a decade ago. They are especially designed for use in number theory, for the purpose of offering the discerning mathematician an alternate representation of the basic, (one exponentiation, one multiplication) algebraic term: $Tc^z,$ where $T, c, z$ represent non negative integers. There were many reasons for their developement. Among them is the fact that in the representation: $Tc^z,$ the variables $T, c, z$ all have the same intrinsic domain, which is: $T=\{0,1,2...\},$ $c=\{0, 1, 2...\},$ $z=\{0, 1, 2...\}.$ I tend to think of math as both a game and a language anyway, so to me, this is kind of like writing: tuna = fish, cobra = fish, zebra = fish. Therefore, I thought it would be fun to see if the variables in the basic term: $Tc^z$ could be operated on in a way that caused them to assume unique domains, so that as a language, both the variables and their domains would more resemble: tuna = fish, cobra = snake, zebra = horse. In other words, since it was never done before, I wanted to re-write the term: $Tc^z$ in a way such that each and every variable had (and was therefore defined by) it's own unique domain, so I wrote: $Tc^z=\left(Tc\right)^{\left(\frac{\frac{z\ln(c)}{\ ln(T)}+1}{\frac{\ln(c)}{\ln(T)}+1}\right)},$ where the domains of the variables are now: $T=\{2,3,4...\},$ $c=\{1,2,3...\},$ $z=\{0,1,2...\}.$ I then divided both sides by $T$ so as to obtain: $\left(\frac{T}{T}\right)c^z=T\left(\frac{c}{T}\rig ht)^{\left(\frac{\frac{z\ln(c)}{\ln(T)}-1}{\frac{\ln(c)}{\ln(T)}-1}\right)},$ and to my surprise, I discovered that I could no longer "cross out" the cancelled $T$'s, nor could I substitute $\frac{c}{c}=1$ for $\frac{T}{T}=1$ unless $z=1.$ This made sense, because if $Tc^z$ represents a composite number, then cancelling the largest possible factor would leave only some prime number: $c^z,$ and since prime numbers don't have exponents, it must be the case that $z=1$ and $c^z=c$. I knew then that I had invented a fundamental algebraic term that was "superior", in that it automatically necessitates the elimination of the exponent when the largest possible factor is cancelled. In other words, my newly invented term was better at representing primality than the other term. I then decided to test my newly invented term on the concept of "co-primality", and to my utter amazement, found that the "Proof Of The Beal Conjecture" practically wrote itself! (That's why I'm quite certain that it's both true and correct, and that it will, in time, be verified.) I named my new invention "cohesive terms", because they retain their cancelled $T$,s and allow them to get "crossed out" if and only if the true and correct value of the exponent $z$ is in place. I then came to the realization that these "cohesive terms" must be perfectly consistent with the rest of mathematics, because logarithms are perfectly consistent with the rest of mathematics. Thus the name "Cohesive Term Theory". You know, I have often been accused of trying to "re-write the fundamentals of mathematics". That's not the case at all! Nothing could be further from the truth! All I wanted was to introduce an alternate way of writing $\left(\frac{T}{T}\right)c^z.$ That's all! You know, ten years ago, the only known way to express: $\left(\frac{T}{T}\right)c^z$ was to write: $c^z,$ which in my view, is both inadequate and unsatisfactory because it implies either that primes have exponents other than unity, or that the cancelled factor was not the largest possible factor. Now, for the first time in the history of mathematics, we have a new and fundamentally different way of expressing: $\left(\frac{T}{T}\right)c^z,$ which is, of course: $T\left(\frac{c}{T}\right)^{\left(\frac{\frac{z\ln( c)}{\ln(T)}-1}{\frac{\ln(c)}{\ln(T)}-1}\right)}.$ I think that's a good thing! It's more variety and possibilities at the fundamental level, so in my estimation, it makes math more fun and less boring (not to mention more accurate and more powerfull). My last thread is a prime example of the kind of venom that is often spewed at me, so know this: I'm not forcing anyone to use, or even look at my invention. You can call me a "troll", "kook", "crank", "crackpot" or any other hatefull and derogatory word of your choosing. I couldn't care less! If you don't like my invention, don't use it! There are, after all, two ways to write a fundamental (one multiplication, one exponentiation) algebraic term.., my way, and the conventional way. You should at least be happy that my invention gives us a choice! Don.
April 4th, 2009, 09:07 PM   #2
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Re: Cohesive Term Theory

Quote:
 Originally Posted by Don Blazys $Tc^z,$ the variables $T, c, z$ all have the same intrinsic domain, which is: $T=\{0,1,2...\},$ $c=\{0, 1, 2...\},$ $z=\{0, 1, 2...\}.$

Quote:
 Originally Posted by Don Blazys in a way such that each and every variable had (and was therefore defined by) it's own unique domain, so I wrote:
Variables don't have domains, functions have domains. And functions aren't defined by their range -- not even if it's {}. So I don't know what you mean.

Quote:
 Originally Posted by Don Blazys and to my surprise, I discovered that I could no longer "cross out" the cancelled $T$'s, nor could I substitute $\frac{c}{c}=1$ for $\frac{T}{T}=1$ unless $z=1.$
If $cT\neq1$ you can *always* substitute c/c for T/T.

Quote:
 Originally Posted by Don Blazys This made sense, because if $Tc^z$ represents a composite number, then cancelling the largest possible common factor
What do you mean "cancelling the largest possible common factor"? Common factor with what?

Quote:
 Originally Posted by Don Blazys I named my new invention "cohesive terms", because they retain their cancelled $T$,s and allow them to get "crossed out" if and only if the true and correct value of the exponent $z$ is in place.
This also makes no sense. Would you give a numerical example?

Quote:
 Originally Posted by Don Blazys You know, ten years ago, the only known way to express: $\left(\frac{T}{T}\right)c^z$ was to write: $c^z,$ which in my view, is both inadequate and unsatisfactory because it implies either that primes have exponents other than unity, or that the cancelled factor was not the largest possible factor.
What makes you say that? These sorts of algebraic manipulations are common, and date back to Napier.

Quote:
 Originally Posted by Don Blazys I think that's a good thing! It's more variety and possibilities at the fundamental level, so in my estimation, it makes math more fun and less boring (not to mention more accurate and more powerfull).

Quote:
 Originally Posted by Don Blazys My last thread is a prime example of the kind of venom that is often spewed at me, so know this: I'm not forcing anyone to use, or even look at my invention. You can call me a "troll", "kook", "crank", "crackpot" or any other hatefull and derogatory word of your choosing. I couldn't care less! If you don't like my invention, don't use it! There are, after all, two ways to write a fundamental (one multiplication, one exponentiation) algebraic term.., my way, and the conventional way. You should at least be happy that my invention gives us a choice!
There are more than just two ways! There are infinitely many -- in particular, the cardinal of the number of representation is aleph_0. Even recognizing these individual forms is difficult: Richardson 1968 shows that the problem of determining whether the equation A = 0 holds is formally undecidable (many common cases are decidable, but not all cases) as long as the expression A is allowed a certain richness (+, -, *, log 2, pi, exp, sin, abs).

April 4th, 2009, 10:01 PM   #3
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Re: Cohesive Term Theory

CRGreathouse said:
Quote:
"Intrinsic" meaning "natural" "dictated by the operations themselves".

CRGreathouse said:
Quote:
 Variables don't have domains.
You are wrong ! Independent variables have "domains", dependent variables "ranges"!

CRGreathouse said:
Quote:
 This also makes no sense. Would you give a numerical example?
z=1, z>1 doesn't allow T's to get crossed out.

Quoting CrGreathouse:
Quote:
 Common factor with what,
Simple typo, I fixed it , but I'm surprised you couldn't figure that out.

CRGreathouse said:
Quote:
 [/There are more than just two ways! There are infinitely many.
[/quote]

Please show me. Write (T/T)c^z in a way other than whats in my post.

April 4th, 2009, 10:28 PM   #4
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Re: Cohesive Term Theory

Quote:
 Originally Posted by Don Blazys Please show me. Write (T/T)c^z in a way other than whats in my post.
(T/T)c^z = (cT)^z / T^z = d^(2z)
for d = sqrt(c).

You may be interested in my thread here:

 April 4th, 2009, 10:43 PM #5 Newbie   Joined: Mar 2009 Posts: 12 Thanks: 0 Re: Cohesive Term Theory I'm done. Please lock this thread! Don.
April 4th, 2009, 10:47 PM   #6
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Re: Cohesive Term Theory

Quote:
 Originally Posted by Don Blazys Using only the three variables T, c and z.
(cT)^z / T^z. But note that T isn't in the original equation c^z, so defining my own terms seemed fair.

It's not hard to come up with rearrangements. Would you like a formal proof that there are infinitely many rearrangements?

 April 4th, 2009, 11:25 PM #7 Newbie   Joined: Mar 2009 Posts: 12 Thanks: 0 Re: Cohesive Term Theory To: CRGreathouse: I believe you. I'm not a professional mathematician, and wasn't aware such esoteric concepts and constructs even exist. My idea is fundamental, but It seems to me that you don't see any beauty or even the slightest modicum of merit in my "cohesive term", so I really think that I'm wasting my time here. Thanks for the good suggestion that you gave me on the proof, but it seems that you can't help me with that anymore either. Don.
April 4th, 2009, 11:57 PM   #8
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Re: Cohesive Term Theory

Quote:
 Originally Posted by Don Blazys I'm not a professional mathematician, and wasn't aware such esoteric concepts and constructs even exist.
There are amazing things in math. I'm glad to have been able to share at least a glimpse of it.

Quote:
 Originally Posted by Don Blazys Thanks for the good suggestion that you gave me on the proof, but it seems that you can't help me with that anymore either.
When you decide to show it to someone, just keep in mind what I told you: be precise, be unambiguous, explain your terms. That way people will be better able to understand. A mathematical proof isn't supposed to leave anything to the reader.

I hope the tips help.

Quote:
 Originally Posted by Don Blazys My idea is fundamental, but It seems to me that you don't see any beauty or even the slightest modicum of merit in my "cohesive term", so I really think that I'm wasting my time here.
It's true; I don't. Perhaps there's nothing there, perhaps I can't see it, or perhaps we just can't communicate properly. I'm focusing on the third possibility here: (hopefully) helping you explain yourself better. That way if you find a more receptive audience you can hopefully show them something of value.

 April 5th, 2009, 01:01 AM #9 Newbie   Joined: Mar 2009 Posts: 12 Thanks: 0 Re: Cohesive Term Theory Thus, my own website is killing me. You are a good friend. Now please close this thread, as I obviously have work to do and won't have much time to reply to others posts. Better yet, just take it down. I will be back when I clear things up. Don.
 April 5th, 2009, 01:41 AM #10 Senior Member   Joined: Oct 2007 From: Chicago Posts: 1,701 Thanks: 3 Re: Cohesive Term Theory Since the OP has requested several times for this thread to be closed, I will do so.

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