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both Japenese, and Americans will realize that problems such as the "Beal Conjecture" and "Fermat's Last Theorem" are the real "joke" because they don't even exist if we represent and eliminate common factors correctly using "Blazys terms", which are the only algebraic terms that actually prevent idiotic "unit common factors" from occurring
There's a nice proof, due to Galileo, that all circles have the same circumference. Do you accept that proposition?
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Here’s a proof of [math] a^x \overset?= T \left(\frac{a}{T}\right)^{\left(\frac{\frac{x\ln(a)}{\ln(T)}-1}{\frac{\ln(a)}{\ln(T)}-1}\right)}[/math]   [math]\log_a b = \frac{\log a}{\log b}[/math], so, f

In this topic, I will be making an extraordinarily "strange claim", almost unbelievable in fact, and it will be up to you, the reader, to decide if the properties of logarithms, strange as they may be

This is a statement that we mathematicians consider totally inaccurate. Don, de L'Hopital's theorem is about limits. How in the world can it be an alternative to "trying" to approach [imath]x=1[/imath

To: Ughaibu,

 

Galileo's "proof that all circles have the same circumference" is quite interesting, and in my opinion, requires "proper interpretation".

 

In Galileo's "cone in bowl" construct, the volume of the "top of the cone" is equal to the volume of the "top of the bowl", while the area of the "base of the cone" is equal to the area of the "base of the top of the bowl".

 

In the end, as the "plane" approaches the very top of the "cone in bowl" configuration, all that is left is the "tip of the cone" which is essentially a "point", and the "outer rim of the bowl", which is essentially a "curved line".

 

Now, the "volume or area" of the resulting "point" can still be viewed as being "equal to" the "volume or area" of the resulting "curved line" in that neither a "point" nor a "curved line" can actually posess any "volume" or "area"!

 

In other words, if the

 

(volume or area of any "point")=0

 

and the

 

(volume or area of any "curved line")=0

 

then with respect to volume and area,

 

(any "point")=(any "curved line" or "circumference")

 

because

 

0=0.

___________________________________________________________________________________

 

That's one way of looking at it... here's another.

 

As the "plane" approaches the very top of the "cone in bowl" configuration, a final "point inside a circle" configuration can never actually be achieved without the plane and cone "losing contact" with each other and therefore "compromising" (rendering meaningless) the entire model or paradigm.

 

In other words, the definition of a "point" as "that which has no part" would actually require the abscence of a cone because clearly, the "tip" of a cone is a "part" of a cone!

 

Well, those are my thoughts on the subject.

 

However, if you "Google search" (Areas Explain Galileo's "Miraculous" Geometry Problem) then you will find some other thoughts on it, as well as some very profound commentary by that trancsendent genius Charles Arthur Mercier.

 

Don.

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Galileo's "proof that all circles have the same circumference" is quite interesting, and in my opinion, requires "proper interpretation"
My point was that this exemplified confusion about dimensions. The limit has no two dimensional component and the one dimensional component, ie the perimeters, were never the same.

Archimedes employed a similar construction in proposition 2 of The Method, without the confusion and in a more radical way. The Method hadn't been recovered at the time of Galileo. Were there only two great minds who thought somewhat alike on this matter? And only one discovered a wonderful analogic method that generated exact results.

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I also find this thread very frustrating.

 

When I first started posting, I never even heard of "LaTex" and was very surprised when my not using it became a "major issue".

 

Rightly or wrongly, I'm afraid many papers and posts are dismissed out of hand if the equations are not formatted with something like LaTex.

 

If you like, Don, I could help you along with the basics and I'm sure you'll pick it up. There is a thread dedicated to practicing LaTex in the test forum. Think of the thread as a place to draw in the sandbox. It's for practicing and nobody is going to mind if you use it as such. I've written you a post there, just follow the link:

 

Latex Practice Ground

 

~modest

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To: Ughaibu,

 

Is "confusion" the same thing as "not knowing", or are they somehow different?

 

You know, as of yet, mathematics doesn't even have a well defined, all encompassing, universally established and accepted definition of that "one dimensional" object called a "line"!

 

Anyway, the relationships between Archimede's "The Method, Proposition 2" and Galileo's "Miraculous Geometry Problem" is such a fascinating topic, with so much room for creative thought and expression, that it threatens to "hijack" this thread, which I summarized in post #75.

 

This topic definitely deserves a thread of it's own!

 

Don.

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Due to numerous requests and "popular demand", I will now re-write the

summary of this topic/thread so that the equations are presented in "LaTex".

 

However, before I do so, I would like to make a few comment's.

 

The title of this topic/thread is in no way "inflated", "exagerated" or "overblown".

 

If students are being taught the wrong way to represent and eliminate common factors,

then that's a very serious matter,

and does indeed constitute a genuine "mathematical emergency"!

 

We simply can not allow nonsensical gibberish to be taught in our schools!

 

:rant_red2: :Angry: :jab: :angryfire: :soapbox: :mad: :rant:

 

A lot of math departments of famous colleges and universities have known about

this result/discovery for years, but have yet to take any remedial action because

they are simply too embarrassed to admit that what they are teaching is wrong!

 

My advise for them is to "get over it", because truth is a very impartial thing.

It's bigger than all of us and doesn't care one iota about our "feelings".

Most importantly, it always wins in the end!

 

Now, here is what each and every student and teacher should know

about the proper representation and elimination of common factors.

____________________________________________________________________________________________________

 

Given the equation:

 

[math]Ta^x+Tb^y=Tc^z[/math],

 

where all the variables are positive integers, how do we eliminate the

common factor [math]T>1[/math] so that all three terms become "co-prime"?

 

Well, these days, students are being taught that we should divide each and every [math]T[/math] by [math]T[/math],

then "cross out" or "cancel out" the [math]T[/math]'s so that they "disappear". Doing so gives us:

 

[math]\left(\frac{T}{T}\right)a^x+\left(\frac{T}{T}\right)b^y=\left(\frac{T}{T}\right)c^z=a^x+b^y=c^z[/math].

 

However, this is wrong because it falsely implies that [math]x, y[/math] and [math]z[/math] can all be greater than 2

when there is no common factor.

 

In reality, and as is stated in the "Beal Conjecture", when there is no common factor,

then we must have a "restriction" on either [math]x, y[/math] or [math]z[/math] so that either:

 

[math]x=[/math] {1, 2}, [math]y=[/math] {1, 2} or [math]z=[/math] {1, 2}

 

So, where did we go wrong?

 

Well, we never actually prevented [math]T=1[/math].

 

You see, preventing [math]T=1[/math] is important, because a true or "non-trivial" common factor [math]T[/math]

is defined as [math]T>1,[/math] and our equations must be consistent with our definitions

if we are to make any real progress and discover the truth.

 

Now, watch what happens if we refuse to prematurely "cross out" the cancelled [math]T[/math]'s,

and instead, re-write the co-prime equation:

 

[math]\left(\frac{T}{T}\right)a^x+\left(\frac{T}{T}\right)b^y=\left(\frac{T}{T}\right)c^z[/math],

 

so that it appears as either:

 

[math]\left(\frac{T}{T}\right)a^x+\left(\frac{T}{T}\right)b^y=T\left(\frac{c}{T}\right)^{\left(\frac{\frac{z\ln©}{\ln(T)}-1}{\frac{\ln©}{\ln(T)}-1}\right)}[/math],

 

or (in it's factored form),

 

[math]\left(\left(\frac{T}{T}\right)a^{(\frac{x}{2})}\right)^2 +\left(\left(\frac{T}{T}\right)b^{(\frac{y}{2})}\right)^2= \left(T \left(\frac{c}{T}\right)^{\left(\frac{\frac{(\frac{z}{2})\ln(c )}{\ln(T)}-1}{\frac{\ln©}{\ln(T)}-1}\right)}\right)^2[/math].

 

Immediately, we find that by substituting just one "Blazys term",

we automatically eliminate any possibility that [math]T=1[/math],

and for the first time in the history of mathematics,

render our equations perfectly consistent with the definition of a non-trivial common factor.

 

Now, take a good close look at the last three equations.

 

Notice that the first one tells us that [math]T=c[/math] is allowable while the next two tell us that

before we can allow [math]T=c[/math], we must first let [math]z=1[/math] and [math]z=2[/math] respectively,

then immediately "cross out" or "cancel out" the logarithms themselves.

Thus, the last three equations must now appear as:

 

[math]\left(\frac{T}{T}\right)a^x+\left(\frac{T}{T}\right)b^y=\left(\frac{T}{T}\right)c^z[/math],

 

[math]\left(\frac{T}{T}\right)a^x+\left(\frac{T}{T}\right)b^y=T\left(\frac{c}{T}\right)[/math],

 

and

 

[math]\left(\left(\frac{T}{T}\right)a^{(\frac{x}{2})}\right)^2 +\left(\left(\frac{T}{T}\right)b^{(\frac{y}{2})}\right)^2= \left(T \left(\frac{c}{T}\right)\right)^2[/math].

 

Now, and only now can we allow [math]T=c[/math], or we can simply "cross out" or "cancel out"

the remaining [math]T[/math]'s, so that the above three equations appear as:

 

[math]a^x+b^y=c^z[/math],

 

[math]a^x=b^y=c[/math]

 

and

 

[math]a^x+b^y=c^2[/math].

 

Notice that the first of the above three equations is a lie because it falsely implies that

if we add together any two co-prime numbers [math]a^x, (x>2)[/math] and [math]b^y,(y>2)[/math],

then we might get a result [math]c^z[/math], such that [math]z>2[/math].

 

The other two equations tell us the truth, which is that

if we add together any two co-prime numbers [math]a^x, (x>2)[/math] and [math]b^y, (y>2)[/math],

then the exponent of [math]c[/math] must be either 1 or 2.

 

Try it yourself! Add together any two co-prime positive integers under the sun,

whose exponents are greater than 2,

and you will find that their sum will always have an exponent that is 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 [math]z=1[/math] and [math] z=2 [/math].

 

Believe it or not, there are some mathematicians who don't think it's possible to "cross out"

or "cancel out" the expressions involving logarithms at [math]z=1[/math] and [math]z=2 [/math].

 

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 that problems such as the "Beal Conjecture" and "Fermat's Last Theorem"

would never have been an issue and would therefore never even have existed

had mankind learned how to properly represent and eliminate common factors to begin with!

 

Thus, it is quite understandable that many in the "math community"

find this irrefutable result to be somewhat embarrassing.

 

It's the true and simple solution to those supposedly "hard" problems!

 

Don.

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Don:

 

I am beginning to see that you have a valid point there. I must admit that the math over there is beyond my skills, but I can see how, when doing a certain type of algebraic menipulation on a function that involves natural logarithms, and a function of the type [math]a^x+b^y=c^z

[/math] something special (and beyond my understanding) happens.

 

BUT…what about the more obvious things?

And I will give a few examples

1. [math]\left(\frac{T}{T}\right)F(x)[/math] if not the same as [math]F(x)[/math] because T might be equal to zero, in which case [math]\frac{T}{T}[/math] is indeterminate !

2. When working with exponents [math]a^x[/math] is not always legal, since if a is a negative number the exponent cannot be a fraction since [math](-1)^{0.5}[/math] has no real answer.

3. [math]\sqrt{x^2}[/math] is not the same as x, since if x is negative, it will become a positive !

 

this list can go on forever, i just came up with a couple examples from my fairly basic knowledge in math, but I believe that this kind of concepts are absolutely essential if you want to understand basic mathematics.

 

How many students are aware of the fundamentals? How many year 12 students who have taken calculus are capable doing simple differentiation using first principals?

How many students know where e, the natural logarithm comes from or are capable of writing and explaining for formula for the series that generates e?

(its [math] \lim_{n\to 0}(1+\frac{1}{n})^n[/math] by the way. just punch into a grahpics calculator [math](1+(\frac{1}{(2^{96})}))^(2^{96}))[/math]:

 

(1+(1/(2^96)))^(2^96) = 2.7182818284590452353602874713355

 

e^1 = 2.7182818284590452353602874713527

(This was fun :wave:)

 

I would even challenge the teachers here to see what percentage of their students know that 'ln' is simple [math] log_e[/math] (!)

 

If I where to use the word emergency, I would use it for this kind of thing. this is not something that concerns mathematicians, but also scientists and engineers (those are the epoeple that have made it possible for this forum to exist through microchips and magnetic drives and stuff like that). They use this level of math all the time and yet a large portion of them are clueless when it comes to actually understanding this level of mathematics.

 

 

:askgoogle:

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To: Logy,

 

Mathematics does not have a "king", but it does have a "queen",

and that queen is "number theory".

 

Number theory is different from other branches of mathematics in that

it deals primarily with the properties of non-negative integers.

 

Most importantly, in number theory, concepts such as "the fundamental theorem of arithmetic",

"common factors", "prime numbers", "perfect numbers", "abundant numbers", etc.

all become utterly meaningless if we allow unity to be both a "multiplier" and a "multiplicand".

 

For example, in number theory, a "perfect number" can be defined as:

"A positive integer such that the sum of its positive proper factors is equal to itself".

Thus, if we allow both "multiplication by unity" and "multiplication of unity",

then [math]6[/math] can be factored as:

 

[math]6=3*2*1*1[/math], where [math]3+2+1+1=7[/math],

 

and the entire concept of a "perfect number" simply collapses!

 

However, if we allow only "multiplication of unity" and strictly prohibit "multiplication by unity",

then [math]6[/math] must be factored as:

 

[math]6=3*2*1[/math], where [math]3+2+1=6[/math],

 

and the concept of a "perfect number" remains intact.

 

You see, the "queen of mathematics" herself is telling us that "multiplication by unity"

is a really dumb idea because it not only destroys important concepts in number theory,

but it also accomplishes absolutely nothing and shouldn't even be considered an "operation"!

 

Now, "coheseve terms" (also known as "Blazys terms") were developed for use in number theory.

They are superior to "conventional" algebraic terms in that they don't allow multiplication by unity,

They are also superior in other ways.

For instance, if we consider the actual meaning of the equation:

 

[math]\frac{T}{T}a^x=T\left(\frac{a}{T}\right)^{\left(\frac{\frac{x\ln(a)}{\ln(T)}-1}{\frac{\ln(a)}{\ln(T)}-1}\right)}[/math]

 

where all the variables represent non-negative integers,

then it is clear that the greatest possible factor [math]T[/math]

was "extracted", then "eliminated" (meaning "cancelled"),

so that both sides must now represent the same "prime number".

 

However, prime numbers do not have exponents, so we are now stuck with

the seemingly impossible task of eliminating the exponent.

 

Again, the "Blazys term" on the right comes to the rescue by telling us that

before we can allow [math]T=a[/math] (which common sense tells us must be allowable),

we must first let [math]x=1[/math], then immediately "cross out" or "cancel out"

the expressions involving logarithms, which leaves us with:

 

[math]\frac{T}{T}a=T\left(\frac{a}{T}\right)[/math],

 

which simplifies to:

 

[math]a=a[/math].

 

So not only do "Blazys terms" provide us with a more meaningfull and accurate

representation of "common factors" (as we saw in post #92),

but they also give us a more meaningfull and accurate representation of factors....period!

 

To put it simply, without "Blazys terms", even our most simple equations will wrongly imply that

eliminating the greatest possible factor from a number does not necessarily result in a prime!

 

You know Logy, you are absolutely right in your assertion that a lot of "high level"

mathematicians don't really understand the underlying principles of the mathematics

that they use, depend on and teach every day.

 

They are like parrots, mynah birds and :turkeytalk: who do everything by rote,

mindlessly regurgitate only what they have been taught,

and never ever bother to actually reason, or think things through for themselves.

 

They are absolutely blind to the fact that "Blazys terms" have the potential to unify

number theory (where multiplication by unity is prohibited), and set theory, algebra,

calculus, etc. (where multiplication by unity is allowed).

 

You are also right in that the "mathematical emergency" does indeed extend far beyond

the idiotic way in which factors and common factors are presented to students.

 

Now here is a question for you.

 

Since it has now been demonstrated that the way in which factors and common factors

are presented to students is every bit as unscientific as "creationism",

shouldn't the courts intervene and put an end to it, just as they did with

the "theory" of "intelligent design"? :protest:

 

Don.

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to: Don

 

the math is getting too time consuming for me. If one day i get into number theory, and am still active in this forum i will revisit this topic.

 

In response to your question, for all i know, you may well be correct. if someone where to come to me 1000 years ago and say that the earth was flat i would dismiss him, but not because his idea is rediculous, rather because it doesnt make sense, how come the people at the bottom don't fall off?

 

only once i see a good explanation would I consider a new idea. (gravity for the above exmple would exlain why people don't fall off) . I believe that this notion is shared among most scientifically oriented people.

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For example, in number theory, a "perfect number" is defined as:

"A number such that the sum of its proper factors is equal to itself".

Thus, if we allow both "multiplication by unity" and "multiplication of unity",

then 6 can be factored as:

 

6=3*2*1*1, where 3+2+1+1=7,

 

and the entire concept of a "perfect number" simply collapses!

 

Why would the number 1 want to be included in the divisors twice? The divisors of 6 are {-6,-3,-2,-1,1,2,3,6}. A perfect number is “a positive integer which is the sum of its proper positive divisors”—proper meaning to exclude the number itself. The proper positive divisors of 6 are then {1,2,3}.

 

I think including 1 twice would assume the divisors are being multiplied, which isn't the case. That 1 • 2 • 3 = 6 (or 1 • 1 • 2 • 3 = 6) is coincidental as far as perfect numbers go. The next perfect number, 28, sums perfectly (28 = 1 + 2 + 4 + 7 + 14), but doesn't multiply perfectly (28 [math]\neq[/math] 1 • 2 • 4 • 7 • 14).

 

So, I'm confused what you did there. And, also in the case of the fundamental theorem of arithmetic, an example such as [math]432 = 2^4 \cdot 3^3[/math] doesn't include the factor 1 because 1 isn't prime. My only guess is that you're thinking [math]432 = 1 \cdot 2^4 \cdot 3^3[/math] wouldn't be unique if [math]432 = 1^2 \cdot 2^4 \cdot 3^3[/math] were allowed. Is that the case? I'm probably way off base.

 

In any case, the multiplicative identity follows as second order logic from the Peano axioms of the natural numbers. So, I don't see how it is at all incompatible with number theory. The wikipedia page on Peano axioms under the arithmetic section has:

It is easy to see that 1 is the multiplicative identity:

a • 1 = a • (S(0)) = a + (a • 0) = a + 0 = a

and using more common logic syntax, axiom #7 under Equivalent axiomatizations:

7. [math]\forall x \in \mathbb{N}. \ x \cdot 1 = x[/math] i.e., one is the identity element for multiplication.

My understanding is that one is included in the natural numbers, so the above explicitly allows 1 x 1 = 1... which I can't reconcile with your statement about number theory not allowing "multiplication by unity and multiplication of unity". :QuestionM :confused:

 

~modest

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To: Modest,

 

Please allow me to both clarify and elaborate.

 

If we are to be consistent in our our use of nomenclature,

then the words "proper factor" must mean the exact same thing as "proper divisor",

because in the case of "proper divisor",

the word "proper" is clearly taken to mean "other than itself".

 

Also, as mathematicians, if there is one thing that we absolutely must agree on,

it's that we must always remain consistent in our mathematical definitions.

 

However, in both the "math community" and on the internet,

there is indeed a whole lot of inconsistency, disagreement and confusion

on what the words "proper factor" and "proper divisor" should mean,

which only adds to the "urgency of the mathematical emergency".

 

Definitions should be both simple and easy to understand,

and since multiplication is "easier" and "more simple" than division,

let us henceforth define "perfect numbers" as I did in post #94,

using "proper factors" rather than "proper divisors".

 

Now, according to the only logically consistent definition of a "proper factor",

the proper factors of [math]28[/math] are:

 

[math]1, 2, 4, 7[/math] and [math]14[/math],

 

and [math]28[/math] is called a "perfect number" because:

 

[math]1+2+4+7+14=28=(1*2*4*7*14)^{\left(\frac{1}{2}\right)}[/math].

 

Note that the proper factors of [math]6[/math] "multiply perfectly" as a "first root",

while the proper factors of [math]28[/math] "multiply perfectly" as a "second root".

If we continue in this manner, then we will find that the proper factors of

the next perfect number, [math]496[/math], also "multiply perfectly", but as a "fourth root",

and that in fact, all perfect numbers "multiply perfectly" as some "Nth root".

(It's one of the "deeper" reasons why these numbers are called "perfect".)

Thus, the argument that I presented in post #94 is not

"coincidental as far as perfect numbers go", but applies throughout.

 

Now, let's take another look at the above equation and note that

the numbers being added together on the far left are an "exact match"

with the numbers being multiplied together on the far right.

 

Clearly, we have a "one to one correspondence" and a "symmetry",

that we dare not break, lest we destroy the foundations of number theory itself.

 

Therefore, as serious seekers of both truth and beauty,

we are now compelled to come up with some reasonable model, paradigm,

or explanation, as to why unity should occur as a factor... only once.

 

After all, considering the incredible depth

of the relationships that abound in number theory,

it would not be unreasonable to suppose

that a reasonable explanation does indeed exist,

and that it is both significant and profound.

 

Do we have any clues? Can we compile some evidence?

 

You betcha and yes we can! :)

 

Consider the following:

 

In number theory, multiplication is strictly and stringently defined as "repeated addition".

Thus, the multiplication: [math]3*5[/math] is viewed as either:

 

[math]3+3+3+3+3[/math] or [math]5+5+5[/math],

 

where the number itself is called the "multiplicand",

and the number of times that it occurs is called the "multiplier".

 

However, applying this definition to the multiplication: [math]1*5[/math] yields either:

 

[math]1+1+1+1+1[/math] or [math]5[/math],

 

and it is now obvious that unity can be defined as a multiplicand,

but not as a multiplier, because clearly, the representation: [math]5[/math]

qualifies as neither a "repeated addition", nor a "repeated" anything!

 

This is absolutely indisputable, utterly irrefutable evidence that

"multiplication and/or division by unity can not be defined.

 

It is so simple that it can be understood by virtually all preschoolers,

yet it is ignored by virtually the entire mathematical community!

 

Thus, the "queen of mathematics" has spoken both clearly and decisively. She said:

"Let this not go in one ear and out the other. Multiplication by unity is undefined!"

 

By contrast, the "Peano" notion of unity being a "multiplicative identity"

or "identity element" is entirely artificial and without foundation, because in truth,

it does nothing more than present us with a rather vaccuous statement

that unity as a factor is "essentially superfluous".

It completely ignores the above direct evidence, which is, in fact, incontravertible,

and in no way addresses the underlying significance of what actually occurs.

 

However, the most compelling and convincing evidence that

"multiplication and or division by unity is undefined" is the "Blazys term" itself,

for at [math]T=1[/math], it actually demonstrates, and clearly shows,

that multiplication and/or division by unity results in division by zero!

 

Thus, given the following four unquestionable facts:

 

(Fact #1): The properties of logarithms exist and are valid.

 

(Fact #2): "Blazys terms" are a logical and therefore unavoidable consequence

of the properties of logarithms.

 

(Fact #3): At [math]T=1[/math], "Blazys terms" demonstrate that

multiplication and/or division by unity results in division by zero.

 

(Fact #4):"Blazys terms" demonstrate that multiplication is non-commutative

with respect to unity, and that unity can exist only as a multiplicand.

 

It logically follows that:

 

(1): The properties of logarithms demonstrate

that multiplication and/or division by unity

is exactly as undefined as "division by zero",

and that unity can exist only as a multiplicand,

so that multiplication must be non-commutative

with respect to unity.

 

and

 

(2): The Peano axiom regarding "multiplicative identities"

is inconsistent with and contradicted by the properties of logarithms.

 

Don.

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To: Modest,

 

Please allow me to both clarify and elaborate.

 

:thumbs_up

 

If we are to be consistent in our our use of nomenclature,

then the words "proper factor" must mean the exact same thing as "proper divisor",

because in the case of "proper divisor",

the word "proper" is clearly taken to mean "other than itself".

 

Also, as mathematicians, if there is one thing that we absolutely must agree on,

it's that we must always remain consistent in our mathematical definitions.

 

However, in both the "math community" and on the internet,

there is indeed a whole lot of inconsistency, disagreement and confusion

on what the words "proper factor" and "proper divisor" should mean,

which only adds to the "urgency of the mathematical emergency".

 

Definitions should be both simple and easy to understand,

and since multiplication is "easier" and "more simple" than division,

let us henceforth define "perfect numbers" as I did in post #94,

using "proper factors" rather than "proper divisors".

 

Well, that's a rather large preamble... and I'm not exactly sure I see the point. Perfect numbers have been defined for over 2,000 years. Factors, for longer. If there's some online confusion and inconsistency... I'm completely out of the loop on that.

 

Now, according to the only logically consistent definition of a "proper factor",

the proper factors of [math]28[/math] are:

 

[math]1, 2, 4, 7[/math] and [math]14[/math],

 

and [math]28[/math] is called a "perfect number" because:

 

[math]1+2+4+7+14=28=(1*2*4*7*14)^{\left(\frac{1}{2}\right)}[/math].

 

Note that the proper factors of [math]6[/math] "multiply perfectly" as a "first root",

while the proper factors of [math]28[/math] "multiply perfectly" as a "second root".

If we continue in this manner, then we will find that the proper factors of

the next perfect number, [math]496[/math], also "multiply perfectly", but as a "fourth root",

and that in fact, all perfect numbers "multiply perfectly" as some "Nth root".

(It's one of the "deeper" reasons why these numbers are called "perfect".)

Thus, the argument that I presented in post #94 is not

"coincidental as far as perfect numbers go", but applies throughout.

 

First, let me say, that's really cool, Don.

 

It appears to be true (or, at least, the first few have no counterexample). I've written a little algorithm to test it up to the 7th perfect number which was successful. The roots were 1,2,4,6,12,16,18. My brain is not working well-enough this Sunday morning to figure why that works... any opinion on that would be much-appreciated.

 

Second, I should say... what you have there is not a definition of a perfect number. It appears I was incorrect using the term "coincidental", but it is, nevertheless, superfluous. A perfect number is an "integer which is the sum of its proper divisors". It's really that simple. You could say the integer (n) is half its divisor function ([math]2n = \sigma(n)[/math]), or equal to its restricted divisor function ([math]n = s(n)[/math]).

But, you cannot say it is "equal to some integer root of the product of its factors". The integer 30 would qualify under that definition as would many others.

[math](1\cdot2\cdot3\cdot5\cdot6\cdot10\cdot15)^{1/3} = 30 \neq (1+2+3+5+6+10+15) [/math]

Moreover, I see nothing about the following equality that prohibits multiplication by unity.

[math](1+2+4+7+14) = (1\cdot2\cdot2\cdot4\cdot7\cdot14)^{1/2} [/math]

I think you want to imply that multiplying the rhs by one would necessitate adding one to the lhs. If that's the case, I don't follow. One is a factor of 28 once.

 

Therefore, as serious seekers of both truth and beauty,

we are now compelled to come up with some reasonable model, paradigm,

or explanation, as to why unity should occur as a factor... only once.

Because [math]1\cdot1\neq28[/math]. One is a factor of 28 because (1 • 28) = 28. You could alternatively write (28 • 1) = 28, but that doesn't make 1 a factor twice any more than it doubles up any of the other factors.

 

Either I'm missing something big or there's nothing about number theory that precludes multiplication by unity. Since multiplication by unity is included in the axioms of natural number arithmetic, I'm pretty confident that it isn't precluded.

 

~modest

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Hey Don. Hey Modest. :wave2: Just thought I'd pop in with some tid-bits I'm somewhat familiar with.

 

When I saw the summing of Perfect factors here, I recalled that the sum of the reciprocals of the proper factors of a Perfect number equals 2 (including the reciprocal of the number itself). Do you guys recall that? Does it shed any light on the discussion? :shrug:

e.g. >> [math]\frac{1}{1}+\frac{1}{2}+\frac{1}{4}+\frac{1}{7}+\frac{1}{14}+\frac{1}{28} = 2[/math]

 

When I saw Modest mention 30 satisfying some of Don's criteria, I recalled that it is one of my "Strange" numbers, which is to say it is abundant by 12. Again, I don't know if any of that forwards this conversation.

 

So I'm not too up on the logarithms but I have been reading all the posts here and maybe some of it will sink in. :help:

 

If we need them, we have threads on Perfect numbers and the Strange numbers. :clue:

 

Perfect Numbers: An Aural Dissertation

 

Strange Numbers

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Ok, my brain has had time to warm up. What Don points out is not a property of perfect numbers and it's not "one of the "deeper" reasons why these numbers are called "perfect".". If fact, every number with an even number of factors will follow Don's little rule. I pulled 30 out of my hat, but consider its factors:

(1 • 30) = 30

(2 • 15) = 30

(3 • 10) = 30

(5 • 6) = 30

Each of these multiply to the number in question. So, the result of multiplying all the factors must be some multiple of the number being factored :thumbs_up

 

A perfect number can't be a perfect square so it must have an even number of factors. Really Don?

 

~modest

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Ok, my brain has had time to warm up. What Don points out is not a property of perfect numbers and it's not "one of the "deeper" reasons why these numbers are called "perfect".". If fact, every number with an even number of factors will follow Don's little rule. I pulled 30 out of my hat, but consider its factors:

(1 • 30) = 30

(2 • 15) = 30

(3 • 10) = 30

(5 • 6) = 30

Each of these multiply to the number in question. So, the result of multiplying all the factors must be some multiple of the number being factored :interesting:

 

A perfect number can't be a perfect square so it must have an even number of factors. Really Don?

 

~modest

 

First & upfront, I'm not at all sure about what Don is saying which is why I'm not offering any opinions on it. Of course as we all know, I don't let little details like that stop me from saying something in response anyway. :coffee_n_pc: I see the triviality of your point though Modest, and you say something further here which again rings my little bell, & that is in regard to squares & perfects. Lost amidst the many diversions in the strange numbers thread is a conjecture I posited and that Craig proved as theorem. Again I don't know if it is shedding light on the topic at hand or casting a cloud on it. :thumbs_up :shrug: :clue: So, for what it's worth, I give you the The Turtle-CraigD Theorem of Odd Powers of Two. :bdayparty:

 

[old]Conjecture: All odd powers of 2 are either a Cube of an odd power of 2, or the sum of a Perfect-Square-multiple of a Perfect number and a Perfect Square. [(Square*Perfect)+Square]

 

The conjecture can be simplified to:

All odd powers of 2 greater than 32 are at least one Perfect-Square-multiple of a Perfect number plus a Perfect Square. ...

 

The rest of Craig's post and the proof is found here. :help:

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and [math]28[/math] is called a "perfect number" because:

 

[math]1+2+4+7+14=28=(1*2*4*7*14)^{\left(\frac{1}{2}\right)}[/math].

 

Note that the proper factors of [math]6[/math] "multiply perfectly" as a "first root",

while the proper factors of [math]28[/math] "multiply perfectly" as a "second root".

If we continue in this manner, then we will find that the proper factors of

the next perfect number, [math]496[/math], also "multiply perfectly", but as a "fourth root",

and that in fact, all perfect numbers "multiply perfectly" as some "Nth root".

(It's one of the "deeper" reasons why these numbers are called "perfect".)

What Don points out is not a property of perfect numbers and it's not "one of the "deeper" reasons why these numbers are called "perfect".". If fact, every number with an even number of factors will follow Don's little rule. I pulled 30 out of my hat, but consider its factors:

(1 • 30) = 30

(2 • 15) = 30

(3 • 10) = 30

(5 • 6) = 30

Each of these multiply to the number in question. So, the result of multiplying all the factors must be some multiple of the number being factored :doh:

I think Modest is correct. :thumbs_up

 

Also, the perfect numbers are given the formula [math]2^{n-1}(2^n -1)[/math], where [math]n[/math] is an element of {2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, ...}, OEIS sequence A000043, the same sequence that gives the Mersenne primes, [math]M_n = 2^n -1[/math]. Therefore, the "N" in the "Nth root" of the product of the proper factors of every perfect number that Don describes will always be exactly 1 less its corresponding [math]n[/math]. For example, the product of the proper factors of the 10th perfect number,

[math]191561942608236107294793378084303638130997321548169216[/math]

is

[math]191561942608236107294793378084303638130997321548169216^{88}[/math]

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