A Mathematical Emergency.

[Nootropic]

To clarify the above, what I'm saying is that the equation is not necessarily incorrect, but what claim it implies is completely ridiculous.

[Don Blazys]

To: Nootropic,

Okay, since you don't know that variables are defined by their domains, I will help you along with the first question: "Which term has the better defined variables?"

Given the "Blazys equation":



where the variables all represent non negative integers, if the terms are considered seperately, then the variables comprising the term on the right have the intrinsic domains:

T={2, 3, 4...}, a={1, 2, 3...} and x={0, 1, 2...}.

Note that no two variables have the same intrinsic domain and definition.

However, the variables comprising the term on the left have the domains:

T={1, 2, 3...}, a={0, 1, 2...} and x={0, 1, 2...}

where the variables "a" and "x" both have the exact same intrinsic domain and thus the exact same definition.

Clearly, the "Blazys term" on the right is comprised of "perfectly defined variables" while the term on the left is comprised of "abysmally defined variables". This makes the term on the right vastly superior to the term on the left.

Also, the term on the right is actually the first and only basic algebraic term (one multiplication, one exponentiation) in the entire history of mathematics to have this property!

That fact alone makes it utterly miraculous and gives it a lot more "clout" than the poorly defined term on the left because in mathematics, constructs that are perfectly defined constitute a higher order of logic than constructs that are poorly defined.

Thus, the term on the right, that I named a "cohesive term", is by far the most frightening animal that has ever been unleashed on the mathematical community, because whatever it indicates must be true for non-negative integers!

It also sheds new light on and brings into question all existing constructs and paradigms that foolishly allow unit common factors to occur, and is thus the source of great embarrassment to many in the math community. Others are simply jealous that they didn't think of it!

By the way, there are also many very good professional mathematicians (including a well known N.A.S.A./ J.P.L scientist) that have endorsed my work and have indicated (both in writing and verbally) that it is both interesting and thought provoking. (I posted a few of their letters on my website (donblazys.com).

Thus far, you have offered absolutely nothing of any substance.

I, on the other hand, am simply allowing the true equation to do my talking for me.
You see, I am actually a very humble person and freely admit that what I say doesn't really matter. However, what the equation says at T=1 matters very much, and sooner or later, the entire math community will have to come to grips with what the irrefutable properties of logarithms are telling us.

That will take more courage than currently exists in the math community.

You have in no way logically refuted, dismissed or dispelled the validity of my equation, but mindlessly persist in telling me that it is somehow "wrong". Trust me, the irrefutable properties of logarithms will never ever imply that something is "ridiculous" unless it really is!

Again, if it is wrong, then point out the error. Otherwise, see if you can muster up the courage to answer the three remaining questions in post #65.

Don.

[ughaibu]

Originally Posted by Don Blazys

Now, if N is a non-negative integer, then the "blunt form":
(0/0)=N
implies that any non-negative integer N multiplied by 0 equals 0.
That's a "reasonable implication" because N*0=0 is a "true statement".

In such a case zero divided by zero would equal the set of all possible numbers. What happens then?

[CraigD]

Originally Posted by Don Blazys

if our variables are to represent non-negative integers, then:

(1) Which term has the better defined variables?

This doesn’t seem to me a meaningful question.

Per the given, all variables are non-negative integers. Because is undefined, and division by zero not permitted in ordinary arithmetic, the additional constraints and . These statements fully define the domain of the expression. It makes no sense to me to ask which terms has the better defined variables, as their definitions apply to them throughout the expression.

Perhaps Don means to consider the domains of 2 separate expressions,

(1)

and

(2)

The domain of expression 1 is unconstrained, other than the given (and unnecessary) requirement that they be non-negative integers. The domain of expression 2 is additionally constrained as above. However, it makes no sense to me to call a more constrained domain “better defined”. For a number to be better defined than another, the other must be in some sense less than fully, explicitly defined. All of the variables above are fully, explicitly defined.

To describe the domain of expression 2 as “perfectly defined”, and that of expression 1 as “abysmally defined”, as Don does in post #72 is informal and, IMHO, inflated and silly.

Originally Posted by Don Blazys

(2) Keeping in mind that unity is not an actual common factor but a "trivial" or "degenerate" common factor, which term is more suitable for representing actual common factors?

That 1 is a factor of every number is a key theorem of ordinary airthmatic. A formal system lacking such a theorem would little resemble arithmetic, and, in short, be weird.

Don appears to be misusing the terms “trivial” and “degenerate”.

Trivial is a relative term meaning, roughly, “not difficult”. It isn’t a formal term. Degeneracy is formal term (see the wikipedia link above), but isn’t applicable to a integer or real valued constant.

Originally Posted by Don Blazys

(3) Keeping in mind that what we do to one side of an identity, we must also do to the other, can we "cross out" the cancelled T's the way we were taught in school?

Although I’ve only a vague idea what Don means by “cross out … the way we were taught in school”, and am aware that many students are taught math very poorly, the following is true:



may be written

Originally Posted by Don Blazys

(4) What occurs at T=1 and what does it imply?

A division by zero occurs. In the usual axiomatic system of arithmetic of real numbers, division by zero is an error condition, implying that the expression has an indeterminate value, rather than one of the usual value of an expression that is an equation, true of false.

Originally Posted by Don Blazys

When answering these questions, please remember that the above equation is absolutely new to mathematics ( I discovered it only a decade ago) so there is, as of yet, no "general consensus" on what it actually means.

There are a infinite number of equations that have never been written. Most are of little interest.



makes for a good exercise in algebra to prove (see Algebraic proof of “Blazys equation”), but despite Don’s claims (eg: that it can be used to prove Fermat’s last theorem and Beal’s conjecture), appears to have no further utility, despite Don’s claims to the contrary, such as

Originally Posted by Don Blazys

It's proper interpretation is perhaps the greatest challenge that the mathematical community has ever faced!

and

Originally Posted by Don Blazys

By the way, there are also many very good professional mathematicians (including a well known N.A.S.A./ J.P.L scientist) that have endorsed my work and have indicated (both in writing and verbally) that it is both interesting and thought provoking. (I posted a few of their letters on my website (donblazys.com).

Also, claims such as these latter should be backed up with links or citation. Simply saying support for them is posted at website, and giving its homepage, is not adequate citation.

The past few post of this thread have largely been an exchange of accusations and denials between Don and Nootropic, such as

Originally Posted by Nootropic

One sign of crackpot mathematics is its implications and you fail to realize the implications your apparent discovery imposes.

Originally Posted by Don Blazys

Also, please back up your claims. If you think that my equation is "crackpot mathematics" then show us where the error lies.

Hypography’s site rules require that we back up our claims. However, Nootropic’s claim is, essentially, that Don’s claims are unsupported, and need to be backed up.

This post from another math forum thread reference to Scott Aaronson’s “Ten Signs a Claimed Mathematical Breakthrough is Wrong” Though Aaronson’s list focuses specifically on the NP complete problem, it’s applicable, I think, to math in general, and to this thread in particular. I think Don would benefit from considering how closely his own writing matches the signs in this list, and attempting to make it match less closely. Justly or unjustly, all experienced math readers apply similar heuristics in determining how much effort to put into attempting to understand a post, paper, or website. If your writing triggers a reader’s “BS detector”, it’s unlikely to be taken seriously by her or him, so ignoring such lists is unwise.

First on Aaronson’s list is

1. The authors don’t use TeX.

Hypography isn’t a publisher, so doesn’t have file contents format requirements such as requiring the use of TeX, but our equivalent of this sign is “the poster doesn’t use LaTeX”. Don, in 3 months of posing, you persist in writing mathematical expressions as difficult to read strings, rather than use hypography’s available rendering features. I’m uncertain why, but don’t like it.

[Don Blazys]

To:Ughaibu

0/0=N does indeed tell us that N can be any non-negative integer.

Therefore, it is called "indeterminate" because if there is no further information on how we got 0/0, then we can't determine it's exact value.

It's only use is in making general statements such as: "One raised to any power equals one," or in mathematical symbols: 1^(0/0)=1^N=1.

However, this thread is not about "indeterminate forms". It's about the proper restrictions that must occur if we are to represent and eliminate common factors correctly.

For instance, if we have the equation:

Ta^x+Tb^y=Tc^z,

where all the variables are positive integers, then how do we eliminate the common factor T so that the equation becomes "co-prime" (Contains no common factor.)?

Well, these days, students are being taught that we should divide each and every T by T, then "cross out" the T's so that they "disappear". Doing so gives us:

(T/T)a^x+(T/T)b^y=(T/T)c^z = a^x+b^y=c^z.

However, this is wrong, because it implies that x, y and z can all be greater than 2 when there is no common factor.

In reality, when there is no common factor, then we must have a "restriction" on either x, y or z so that either:

x={1,2}, y={1,2} or z={1,2}.

Where did we go wrong? Well, we never actually prevented T=1, did we? Preventing T=1 is important, because a true or "non-trivial" common factor is defined as T>1.

Now, watch what happens if we refuse to "cross out" the T's "prematurely" and re-write the co-prime equation:

(T/T)a^x+(T/T)b^y=(T/T)c^z

so that it appears as either:

(T/T)a^x+(T/T)b^y=T(c/T)^((zln(c)/(ln(T))-1)/(ln(c)/(ln(T))-1)),

or (in it's factored form):

((T/T)a^(x/2))^2+((T/T)b^(y/2))^2=(T(c/T)^(((z/2)ln(c)/(ln(T))-1)/(ln(c)/(ln(T))-1)))^2.

Immediately we find that by substituting just one "Blazys term", we eliminated any possibility that T=1. Now, take a good close look at the last three equations. Notice that the first one tells us that T=c is allowable while the next two tell us that before we can allow T=c, we must first let z=1 and z=2, then immediately "cross out" the logarithms themselves. Thus, the last three equations now appear as:

(T/T)a^x+(T/T)b^y=(T/T)c^z,

(T/T)a^x+(T/T)b^y=T(c/T)

and

((T/T)a^(x/2))^2+((T/T)b^(y/2))^2=(T(c/T))^2.

Now, and only now can we allow T=c, or we can simply "cross out" the remaining T's so that the above three equations appear as:

a^x+b^y=c^z,

a^x+b^y=c

and

a^x+b^y=c^2.

Notice that the first of the above three equations is a lie because it implies that if we add together any two co-prime numbers a^x, (x>2) and b^y, (y>2), we might get a third number c^z where z>2.

The other two equations tell us the truth, which is that if we add together any two co-prime numbers a^x, (x>2) and b^y, (y>2), then the exponent of c must be either 1 or 2.

Try it yourself! Add together any two co-prime positive integers under the sun with exponents greater than 2 and you will find that their sum will always have an exponent of either 1 or 2.

Most importantly, notice that "indeterminate forms" such as 0/0 are never ever encountered if we do the algebra correctly and "cross out" or "cancel out" the expressions involving logarithms the very moment that we let z=1 and z=2.

Believe it or not, there are some mathematicians who don't think it's possible to "cross out" or "cancel out" the logarithms at z=1 and z=2.

I think that they are mistaken.

I think that conjuring up "indeterminate forms" that don't even exist is just plain silly.

Not only is this the correct way to represent and eliminate "common factors", but it also shows us that problems such as the "Beal Conjecture" and "Fermat's Last Theorem" would never have existed had mankind learned how to properly represent and eliminate common factors to begin with! Thus, it's quite understandable that many in the math community find this irrefutable result to be "embarrassing".

It is the solution to those supposedly "hard" problems!

Don.

[Don Blazys]

To:Craig D.

When discussing the domains of the variables in the equation, I did say that the terms should be considered or taken seperately. However, we are not discussing the definitions of numbers, but of variables.

In other words, we are discussing the fact that any variable is defined by it's domain, and if two different variables have the same domain, then they have, in essence, the same definition.

The concept is very simple. If we define the variables a and x as:

a={0,1,2...} and x={0,1,2...},

then a and x mean the exact same thing. However if we define a and x as:

a={1,2,3...} and x={0,1,2...},

then clearly, a and x mean two different things!

To put it in yet another way, if we let the symbols:

"a" or "{1,2,3...}" represent the word "cat", and

"x" or "{0,1,2...}" represent the word "dog",

then without "Blazys terms", we would forever be forced to conclude that cats are dogs because all we would ever have is:

a={0,1,2...} and x={0,1,2...}.

So you see, by the proper restriction of domains, "Blazys terms" provide us with a great improvement in definition!

That's a good thing, because only an idiot would call a cat a dog!

I am making no claims whatsoever, but simply allowing the equations:

(T/T)a^x=T(a/T)^((xln(a)/(ln(T))-1)/(ln(a)/(ln(T))-1))

and (it's factored form):

((T/T)a^(x/2))^2=(T(a/T)^(((x/2)ln(a)/(ln(T))-1)/(ln(a)/(ln(T))-1)))^2

to speak for themselves. They, not I say that letting T=1 is not possible. In other words, they, not I say that common factors, when they are allowed to become "trivial", clearly result in division by zero!

How would you "cross out" the T's in the above equations?
(By "crossing out", I mean "cancelling out" so that the cancelled T's "disappear".)

Common factors are not unity, period! In fact, the phrase "a common factor of unity" is actually a "misnomer" in that it is supposed to "mean" that no common factor exists! Therefore, eliminating the possibility of multiplication by unity is not "wierd" but a basic concept of number theory! I'm simply the first to write an algebraic term that reflects that concept perfectly.

You see, in number theory, unity is never viewed as a "multiplier" but as a "multiplicand". The reason for this is that the fundamental theorem of arithmetic tells us that every factorization is unique. Thus, if we allow multiplication by unity, then the number 6=1*2*3 would also be "factorable" as 1*1*2*3, 1*1*1*2*3 and so on. The sums of those factors would then be 6, 7, 8 and so on, and concepts such as "perfect numbers", "abundant numbers" and so on, would all collapse!

"Blazys terms" were designed for use in number theory. They are the first and only algebraic terms that don't allow trivial common factors to creep in to our equations. Moreover, they prevent the loss of cancelled common factors, allow us to develop incredible one and two term prime counting functions and present us with an entirely new and more rigorous form of calculus. To say that there are an infinite number of ways to write an elementary expression such as a "Blazys term" is simply ludicrous. There is, essentially, only one way to write a "Blazys term" (where T>1). All the rest are simply "variations".

None of the letters on my website are forgeries! Sorry, but I only began using a computer recently, so I don't know how to set up "links", nor do I know how to write in "LaTex" or put up "Smilies".

Euler, Fermat and Gauss didn't use LaTex either.

It doesn't make me wrong, just "old fashioned".

Don.

[Don Blazys]

To: Craig D,

I just now noticed that a few posts ago, you took the trouble to edit my equation so that it now appears in LaTex.

That's very kind of you!

Very kind!

Regardless of whether or not you agree with me, I, like any other Hypographer, strive to make my posts as entertaining as possible... and that sure helped!

Thanks!

Don.

[logy]

After reading through a few pages of this topic, which seemed quite interesting to begin with, I have noticed a few recurring patterns

1. A gentleman called Don Blazys keep presenting complex arguments in a form that is very difficult to read, even for math professionals !
2. He is also making some claim about and is using a complicated math identity and is totally unrelated to that claim in order to make it look fancy.
3. The main reason that most people can't see through this, is due to the combination of the fact that the arguments are quite complicated, and the fact that they are being presented in an unreadable way !
4. both Craig and Qfwfq have been more then accommodating, to the point of reformatting the arguments into a more readable format, as well as entertaining the notion that [math\frac{T}{T} a^x [/math] is different somehow from simply .

I’m not a professional mathematician but I do have a grasp of some fundamental concepts and so:



is the same as


is the same as


I have intentionally used the term 'some function' because even though it appears to be a fascinating identity (to me, a non professional), it has absolutely nothing to do with .

I do agree with Craig that it should not be canceled out **automatically** because that leads to lack of understanding and ultimately to stupid mistakes, but rather should be canceled out with care, understanding that , since dividing a number by itself gives you 1, and multiplying a function by one does not change the result

I recently had a long discussion with a mathematics professor who teaches year 1 engineering students and he claimed that his students tend to make fatal errors due to lack of understanding specifically he referred to being simplified to , which is incorrect since not the same as .

Do you agree guys? if you have anything to say, please do

(umm…unless your name is Don Blazys, in which case, please use LATEX )

[CraigD]

Originally Posted by logy

After reading through a few pages of this topic, which seemed quite interesting to begin with, I have noticed a few recurring patterns

1. A gentleman called Don Blazys keep presenting complex arguments in a form that is very difficult to read, even for math professionals !
2. He is also making some claim about and is using a complicated math identity and is totally unrelated to that claim in order to make it look fancy.
3. The main reason that most people can't see through this, is due to the combination of the fact that the arguments are quite complicated, and the fact that they are being presented in an unreadable way !
4. both Craig and Qfwfq have been more then accommodating, to the point of reformatting the arguments into a more readable format, as well as entertaining the notion that is different somehow from simply .
...
Do you agree guys? if you have anything to say, please do

I agree – oh, and a belated welcome to hypography, logy!

Originally Posted by Don Blazys

Sorry, but I only began using a computer recently, so I don't know how to set up "links", nor do I know how to write in "LaTex" or put up "Smilies".

To create links, enclose the URL of the page to which you wish to link with [url] and [/url]. For example,
[url=http://hypography.com]Hypography Science Forums[/url]
will produce this link: Hypography Science Forums. To have your link display text other than the URL’s title, use something like
[url=http://hypography.com]hypography’s main page [/url]
, which display as hypography’s main page .

To enter a smiley, type :). For more exotic ones than a simple , like , click the “more” link on the “smiley” panel on the “Reply to Thread” page, and select from the page that pops up.

Originally Posted by Don Blazys

Euler, Fermat and Gauss didn't use LaTex either.

It doesn't make me wrong, just "old fashioned".

Euler, Fermat and Gauss didn't use LaTeX because, when they wrote, it didn’t exists. Had it, they almost certainly would have.

They wrote in the best and most accepted medium of their day, pen on paper. When writing for others, they wrote neatly, so that their writing could be understood with a minimum of effort spent interpreting their glyphs and notation. They didn’t writing in stings of horizontally separated characters, press cuneiform marks into clay tablets, etc., because this would have made their writing unnecessarily difficult to read.

Pen/pencil on paper is still an acceptable medium. To transmit it electronically, it’s necessary to capture a facsimile of it with a device such as a scanner or digital camera. These image files can be uploaded to your hypography “photos” gallery or any of many free image hosting websites. Because this process is labor and time-consuming, and the resulting images usually less readable than a LaTeX-rendered equivalent, most math writers prefer to use the LaTeX markup language, either transcribing their paper notation into it, or working in it directly, or using editors such as this online equation editor.

Like most markup languages, the LaTeX math package is easy to learn. There are many online references and tutorials, including this hypography thread, and this one. You may see examples of the LaTeX used in any post by clicking on its “quote” button. And, finally, any thread with questions or request for instruction on using LaTeX or other site features will be gleefully answered by our members.

If you persist in refusing to learn conventional online writing techniques, Don, you’re likely to be taken un-seriously or worse. Because most readers are aware how easy they are to learn, not using them leads one to wonder why someone should bother reading someone who is unable to learn them, or suspect that you are purposefully avoiding learning or using them in order to make yourself difficult to understand. I don’t believe you intend the latter, nor are unable to learn to these techniques, so strongly encourage you to learn and use them.

[ughaibu]

Originally Posted by CraigD

the LaTeX math package is easy to learn

Those using a japanese keyboard might take issue with this, when I tried experimenting with Latex I found very little correspondence between the instructions and the output.