A Mathematical Emergency.

[modest]

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. :smilingsun: :shrug: So, for what it's worth, I give you the The Turtle-CraigD Theorem of Odd Powers of Two. :)

 

Yeah, I recall that... vaguely. I'll have to look it over, but I don't know the particulars of why a perfect number cannot be a perfect square. It's purportedly proven here for both even and odd perfects:

http://www.goshen.edu/~dhousman/ugresearch/Soe%202001.doc

If this is true (and wikipedia says it is) then all perfect numbers should have an even number of factors by virtue of not being a perfect square and should likewise follow the rule that Craig proved in the last post (which I think may apply to even perfects only).

 

In any case, I don't see how any of this could possibly mean multiplication by unity is prohibited by number theory.

 

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]

 

 

Not enough +rep to go around :(

 

~modest

[Don Blazys]

To: Modest, Turtle and Craig D,

 

You are all fine mathematicians.

 

The "force" is with you my friends.

 

Now, if you could be turned to the "Blazys side",

 

then you would all be powerfull allies! :rolleyes:

 

All kidding aside, your responses are numerous, and your issues are many,

so I will address them in a nice orderly fashion, that is, one at a time,

in the order that they were made.

 

Now, the very first issue, brought up by Modest, is that he is, in his own words,

"out of the loop" on the inconsistency, disagreement and confusion

that exists in both the math community and on the internet

as to what the words "proper factor" and "proper divisor" should mean.

 

Therefore, I invite everyone to "Google search" the words

"proper factor" and "proper divisor", and verify my observations that:

 

(1) "Wolfram Mathworld" defines "proper factor" differently from "WikiAnswers".

 

(2) The "Wolfram Mathworld" article on "proper divisor" points out the

fact that "proper divisor" is often defined as excluding both -1 and 1,

and unequivocally states that confusion and disagreement on the

meaning of the words "proper divisor" do indeed exist.

 

(3) In "Ask Dr. Math", Dr. Tom and Dr. Peterson disagree on what constitutes a

"proper factor".

 

(4) In "Ask Dr. Math", Dr Greenie also states that there is much confusion

among mathematicians on the meanings of the words "proper factor" and "proper divisor",

then contradicts himself when he says "you should find no disagreement among

mathematicians that the "proper divisors" of 8 are 1, 2 and 4."

 

So, my first question to you, my friends, is this.

 

Is there inconsistency, disagreement and confusion in both the "math community",

and on the internet, as to what the words "proper factor" and "proper divisor"

should mean?

 

This is a simple yes or no question.

 

Let's stay "on topic" and answer just this one question

before we move on to the other issues.

 

Don.

[modest]

Hello Don, I think you've accidentally missed all major points of contention and criticism.

 

I must first correct you in that I am not a mathematician. The extent of my education in math was calc2 in college 10 years ago, and I've not kept current with any publications in the area of study since.

 

Secondly, I believe successful communication relies on a common understanding of the terms used to communicate.

 

I don't mind if you use the term divisor or factor, or "proper divisor" or "proper factor" so long as you make clear what it is you are referring to. In specific cases, "proper factor" can be defined as to exclude the number 1 and "divisor" can be defined as to include negatives.

 

To define the restricted divisor function (i.e. aliquot sum) avoiding these two issues it can be defined as "the sum of all positive proper divisors" and a perfect number can be defined as "a positive integer which is the sum of its proper positive divisors”

 

The first thing I did when responding to your topic of perfect numbers was to define my terms:

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 later gave two more definitions of "perfect number". It is a number which is equal to its restricted divisor function, s(n), (i.e. aliquot sum). Or, a number which is equal to half its divisor function, [math]\sigma(n)[/math]. I will now give examples to clarify:

 

The number 12 is not perfect because,

[math]s(12) = 1+2+3+4+6 = 16[/math]

[math]s(12) \neq 12[/math]

The number 28 is perfect because,

[math]s(28) = 1+2+ 4+7+14 = 28[/math]

[math]s(28) = 28[/math]

 

If it is at all unclear how I am using these terms then I will break it down further. If you wish to use terms differently (or use different terms altogether) then I encourage you to define them.

 

~modest

[CraigD]

Is there inconsistency, disagreement and confusion in both the "math community",and on the internet, as to what the words "proper factor" and "proper divisor" should mean?

Yes.

 

However, this is true of practically any words used in any discipline, if the “community” considered is sufficiently broad. “The math community” and “the internet” are a very broad communities.

 

Among recognized mathematicians (eg: people with PhDs in Math and related disciplines), I think there’s much inconsistency, but little disagreement or confusion about the meaning of any phrase, because mathematicians try to carefully define the words they use in a particular context, rather than relying on a pre-defined usual and traditional meanings.

 

Thus, while the usual meaning of “proper divisor of n” and “proper factor of n” are “an integer d such that 0 < d < n and the remainder of n ÷ d is 0” and “an integer d such that 1 < d < n and the remainder of n ÷ d is 0”, as Modest notes in post #105, and Don observes by finding a contradiction in an “Ask Dr. Math” webpage, variations in meaning are permitted and not uncommon.

 

The adjective “proper” usually means that the meaning of the noun following it is restricted in some special way. The Wolfram Mathworld entry for “proper” gives the terse definition “in general, the opposite of trivial.” I’d define it more liberally as “used in the way I’m using it right here”. So, when one encounters a phrase like “proper divisor”, the “proper” is a clue that one must search out a precise definition of the phrase.

 

Personally, I think math is most easily read when written with the smallest vocabulary. The standardization of and widespread familiarity of math readers with real and pseudo programming languages has, I think, greatly aided communication, because it’s possible to exactly define functions and their output as (usually) short programs. For example, the perfect numbers are defined by the following MUMPS program:

f  s N=N+1,S=0 x "f D=1:1:N-1 s:N#D=0 S=S+D" q:N=S

which sets the variable N to the next perfect number greater than its initial value. Because the interpreters of such program are themselves explicitly defined programs, they can’t be misinterpreted due to human subjectivity.

[Don Blazys]

To: Modest,

 

When you wrote that I "accidentally missed all major points of contention and critisism",

you forgot that I am addressing the many points that were made, one at a time,

in the order that they were made, because that's the best way to avoid further confusion.

Please be patient. I will get to each and every point, I promise.

 

Anyway, now that we have established that the "math community"

is in a state of confusion as to whether or not unity constitutes a

"proper factor" or "proper divisor", let's move on to the next point,

which is one of those "points of contention".

 

Quoting Modest:

One is a factor of 28 once.

 

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.

 

Well, I agree that [math]1[/math] is a factor of [math]28[/math] only once.

But, here is the real question:

 

If we have the multiplication:

 

[math]1*28[/math],

 

then did we begin with [math]1[/math] and increase it by a factor of [math]28[/math],

 

or did we begin with [math]28[/math] and "increase" it by a factor of [math]1[/math]?

 

You see, while it's clearly possible to increase unity by a factor of twenty-eight,

it's simply not possible to "increase" anything by a factor of unity,

so "multiplication by unity" can't possibly exist, and something that

can't possibly exist, can't possibly be defined!

 

Thus, if unity occurs in a factorization only once,

then it is possible to interpret that factorization "meaningfully",

because unity can then be viewed as "strictly a multiplicand".

However, if unity occurs in a factorization twice,

then it is no longer possible to interpret that factorization "meaningfully",

because unity would then have to be viewed as both a "multiplicand",

and a "multiplier".

 

In other words, if unity occured in a factorization twice,

then we would, in fact, be lying, because the "multiplication":

 

[math]1*1[/math]

 

can only be interpreted as:

 

"We began with [math]1[/math], then "increased" it by a factor of [math]1[/math] ".

 

As I mentioned in post #97, in number theory, multiplication is strictly and stringently

defined as "repeated addition", so that 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".

 

Now, if we apply this definition to the multiplication: [math]1*28[/math], then we get either:

 

[math]1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1[/math] or [math]28[/math],

 

Notice that the representation:

 

[math]28[/math]

 

does not qualify as a "repeated addition", or, for that matter, a "repeated" anything!

 

This tells us unequivocally that unity is definable only as an operand.

In other words, "multiplication of unity" can be defined as a "repeated addition",

but "multiplication by unity" can not be defined as a "repeated addition".

 

Applying this "repeated addition" definition of multiplication to the multiplication: [math]1*1[/math] yields:

 

[math]1[/math],

 

and it is now painfully obvious that the representation: [math]1[/math]

does not "repeat", and that the representation: [math]1*1[/math]

has nothing whatsoever to do with multiplication as it is defined in number theory!

 

Now, the only reason that "multiplication by unity" has been tolerated for so long,

is because up until a decade ago, mathematicians had no way of precluding it.

 

In other words, until I invented "Blazys terms" about ten years ago,

mathematicians had no idea as to how to write an algebraic term

that is perfectly consistent with the actual definition of multiplication

as it applies to unity!

 

"Blazys terms" verify the above observation that "multiplication by unity" is undefined,

and since they are nothing more than a logical and therefore unavoidable consequence

of the properties of logarithms, it is clear that any notions whatsoever regarding

"multiplicative identities" or "identity elements" can not possibly qualify as "axioms",

because any such notions are clearly inconsistent with both the definition of multiplication,

and the properties of logarithms.

 

We have already shown that the "math community" has a very poor understanding of unity.

If it can't even agree on whether or not unity constitutes a "proper factor",

or even a "proper divisor", then how can it be trusted with the subtle nuances of

establishing it's axioms !?

 

When it comes to science, we must not be like lemmings, blindly following each other off a cliff!

We must be men (and women) who actually use our God given ability to reason!

 

Don.

[Don Blazys]

To: Craig D,

 

Thanks for the straight answer!

 

All I can add to that is that definitions of fundamental concepts must be completely

unambiguous and easy to understand if a subject such as mathematics is to garner

the publics interest and thereby greatly accelerate it's progress.

 

Don.

[Turtle]

...

But, here is the real question:

 

If we have the multiplication:

 

[math]1*28[/math],

 

then did we begin with [math]1[/math] and increase it by a factor of [math]28[/math],

 

or did we begin with [math]28[/math] and "increase" it by a factor of [math]1[/math]?

 

You see, while it's clearly possible to increase unity by a factor of twenty-eight,

it's simply not possible to "increase" anything by a factor of unity,

so "multiplication by unity" can't possibly exist, and something that

can't possibly exist, can't possibly be defined!

 

Don.

 

Hi Don. Allow me some wordsmithing if you will here. Substitute your "increase" with "make arrangements of " and consider that you have actually piles of beans in the number that the numerals represent. In this case, the 1 * 28 has the expression " make 1 arrangement of 28" or "make 28 arrangements of 1", or vice versa if you write 28*1. The product is of course the sum of all arrangements then made.

 

I have mused over the apparent "error" in leaving out factors, including but not exclusive to 1. For example strictly following the rule for Perfect Numbers, i have argued that 1 is Perfect. Problem is, down the road it makes for all manner of inconsistencies as it makes 1 and not 6 the first Perfect. It's a fine philosophical debate, but there is good reason for settled conventions. A non-one example is the question of whether 16 is abundant or deficient, and it depends on if you count 4 twice as a factor. It is still not clear to me what repurcussions follow from counting it twice, but 16 is not alone in this as other squares also change their stripe on this condition.

 

With all the interest and discussion on your ideas, does anyone feel there is room to call it a "different algebra" and simply go about seeing what complications arise based on it?

 

That's a rap. :)

[modest]

To: Modest,

 

When you wrote that I "accidentally missed all major points of contention and critisism",

you forgot that I am addressing the many points that were made, one at a time,

in the order that they were made, because that's the best way to avoid further confusion.

Please be patient. I will get to each and every point, I promise.

 

I do appreciate that.

 

It’s all too often on forums such as this that valid rebuttals are made for some given argument only for the person making the argument to move the goalpost ignoring the scope of the rebuttal entirely. It’s an informal logical fallacy that gets a lot of play around here which has no doubt made me somewhat skeptical of posts that take the form “In time we’ll get to the main issue raised, but first...”. I appreciate your assurances that this is not the case and apologize for assuming otherwise.

 

I think you're bringing up a lot of interesting and worthwhile things, but I don't see anything addressing my original question from a week ago. I'm referring in particular to this:

 

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!

 

It reads to me like you are saying that if we are allowed to factor 28 as (1 • 1 • 2 • 2 • 7) or [math](1^2 \cdot 2^2 \cdot 7) = 28[/math] rather than the normal factorization [math](2^2 \cdot 7) = 28[/math] then its positive proper divisors would be {1,1,2,4,7,14} rather than the normal {1,2,4,7,14} which then wouldn't work for describing a perfect number. I don’t understand how you get that set of divisors. How do you get from the factorization:

(1 • 1 • 2 • 2 • 7) = 28

to the summing of positive proper divisors:

s(28) = 1 + 1 + 2 + 4 + 7 + 14 = 29

 

I fully accept that I may be misunderstanding something about factoring that makes the above obvious, but as it stands, it appears simply to be mistaken.

 

If I were to look for the positive proper divisors of a number in order to check if it is perfect, I would find all positive integers which divide the number without leaving a remainder. Those numbers would make a set from which I would exclude the number itself and then sum the set.

 

As this relates to factorization, if I were to decompose the number 28 into a factorization:

[math](2^2 \cdot 7) = 28[/math]

and I wanted to relate that to the positive divisors of 28, then I would say a divisor of 28 (D) is:

[math]D = (a^x \cdot b^y)[/math]

where a=2, b=7, x={0,1,2}, and y={0,1}. This would mean any of the following are divisors of 28 given the factorization above:

[math]2^0 \cdot 7^0 = 1[/math]

[math]2^0 \cdot 7^1 = 7[/math]

[math]2^1 \cdot 7^0 = 2[/math]

[math]2^1 \cdot 7^1 = 14[/math]

[math]2^2 \cdot 7^0 = 4[/math]

[math]2^2 \cdot 7^1 = 28[/math]

 

I don’t know how normal it is to consider (1 • 1 • 2 • 2 • 7) a possible factorization of 28. It seems rather redundant, but that does not mean it is wrong or impossible or inconsistent with number theory. Relating this factorization,

[math](1^2 \cdot 2^2 \cdot 7) = 28[/math]

to the divisors of 28, I would say,

[math]D = (a^x \cdot b^y \cdot c^z)[/math]

where a=1, b=2, c=7, x={0,1,2}, y={0,1,2}, and z={0,1}. This would seem to mean that the result of any of the following would be possible divisors of 28 given the factorization (1 • 1 • 2 • 2 • 7)=28:

[math]1^0 \cdot 2^0 \cdot 7^0 = 1[/math]

[math]1^0 \cdot 2^0 \cdot 7^1 = 7 [/math]

[math]1^0 \cdot 2^1 \cdot 7^0 = 2 [/math]

[math]1^0 \cdot 2^1 \cdot 7^1 = 14[/math]

[math]1^0 \cdot 2^2 \cdot 7^0 = 4[/math]

[math]1^0 \cdot 2^2 \cdot 7^1 = 28[/math]

[math]1^1 \cdot 2^0 \cdot 7^0 = 1[/math]

[math]1^1 \cdot 2^0 \cdot 7^1 = 7[/math]

[math]1^1 \cdot 2^1 \cdot 7^0 = 2[/math]

[math]1^1 \cdot 2^1 \cdot 7^1 = 14[/math]

[math]1^1 \cdot 2^2 \cdot 7^0 = 4[/math]

[math]1^1 \cdot 2^2 \cdot 7^1 = 28[/math]

[math]1^2 \cdot 2^0 \cdot 7^0 = 1[/math]

[math]1^2 \cdot 2^0 \cdot 7^1 = 7[/math]

[math]1^2 \cdot 2^1 \cdot 7^0 = 2[/math]

[math]1^2 \cdot 2^1 \cdot 7^1 = 14[/math]

[math]1^2 \cdot 2^2 \cdot 7^0 = 4[/math]

[math]1^2 \cdot 2^2 \cdot 7^1 = 28[/math]

Since we know D={1,2,4,7,14,28} then we know none of the above are wrong. So, I just don't see how the factorization,

(1 • 1 • 2 • 2 • 7) = 28

implies the restricted divisor function:

s(28) = 1 + 1 + 2 + 4 + 7 + 14 = 29

How did you get from one to the other? What happened between those two steps?

 

~modest

[Don Blazys]

Hi Turtle,

 

And .

 

Now, when you wrote:

 

Quoting Turtle:

Substitute your "increase" with "make arrangements of "

and consider that you have actually piles of beans in the number that the

numerals represent. In this case, the 1 * 28 has the expression

" make 1 arrangement of 28" or "make 28 arrangements of 1",

or vice versa if you write 28*1. The product is of course

the sum of all arrangements then made.

 

you gave me an idea as to how the operation of multiplication can be

rigorously defined in terms of "beans"!

 

I like beans! They are good for my heart!

The more I eat, the more I'm :smart:

And although our lives may not be worth

a hill of beans in this crazy world,

this is our hill, and these are our beans!

 

:eek_big: :eek: :eek2:

 

There aren't enough "bean counters" here to warrant "bean shaped smilies"!

 

That's okay, I will use "O" instead. After all, it "kinda" looks like a bean.

 

Alrighty then!

 

In this this demonstration, we will use "repeated columns of beans"

to rigorously define multiplication as "repeated addition".

Now, if we call the number of beans in a column the "multiplicand",

and the number of columns the "multiplier",

then the multiplication [math]3*5[/math] can be shown as either:

 

3+3+3+3+3

O O O O O

O O O O O

O O O O O

 

or

 

5+5+5

O O O

O O O

O O O

O O O

O O O

 

Notice that the columns "repeat" in two ways,

as "five columns of three beans"

and as "three columns of five beans".

 

However, if we apply this definition

to the multiplication [math]1*5[/math],

then all we have is either:

 

 

1+1+1+1+1

O O O O O

 

or

 

5

O

O

O

O

O

 

where it is clear that

the columns "repeat" in only one way,

as "five columns of one bean".

 

The "one column of five beans" does not "repeat".

 

It occurs only once.

 

Thus, unity can be defined as a multiplicand, but not as a multiplier,

and "multiplication of unity" can be defined as the "repeated addition of unity"

while "multiplication by unity" can not be defined as the "repeated addition of five",

or, for that matter, as the "repeated addition" of anything!

 

Multiplication by unity is simply "undefined"!

 

Leave it to you to inspire this most entertaining simplification!

 

Don.

[Turtle]

Hi Turtle,

And .

...

That's okay, I will use "O" instead. After all, it "kinda" looks like a bean.

 

In this this demonstration, we will use "repeated columns of beans"

in order to rigorously define multiplication as "repeated addition".

Now, if we call the number of beans in a column the "multiplicand",

and the number of times those columns occur the "multiplier",

then the multiplication [math]3*5[/math] can be shown as either:

 

3+3+3+3+3

O,O,O,O,O

O,O,O,O,O

O,O,O,O,O

 

or

 

5+5+5

O,O,O

O,O,O

O,O,O

O,O,O

O,O,O

 

However, if we apply this definition to the multiplication [math]1*5[/math], all we have is either:

 

1+1+1+1+1

O,O,O,O,O

 

or

 

5

O

O

O

O

O

 

where it is clear that the [math]1[/math] column of [math]5[/math] beans does not "repeat" as do the [math]5[/math] columns of [math]1[/math] bean.

 

Thus, unity can be defined as a multiplicand, but not as a multiplier,

and "multiplication of unity" can be defined as "repeated addition of unity"

while "multiplication by unity" can not be defined as "repeated addition of five."

 

Leave it to you to inspire this most entertaining simplification!

 

Don.

 

Roger the welcome. This is what the bean-counters pay me the big bucks for. :hihi:

 

So to the worsmithery! The mistake here I think is that you have taken my generalized principle term "arrangement" and substituted a specific example term "column". You are correct that one arrangement is a column and the other not, but that is because the other arrangment of five beans is a row. Both arrangments contain a total of 5 beans and one might make a line of 5 separate beans on any compass heading with no effect on their quantity.

 

Imagine now that you have an endless supply of glass beakers in which to place arrangements of beans, which is to say amounts of beans as they will self-arrange physically in the beaker. Imagine also an endless supply of beans.

 

Let'c count out 5 beans and put them all in a single beaker. This is multiplication by unity. One beaker of 5 beans. Now we can proceed to take all but 1 of those beans from the beaker and set about seeing if we can put those beans taken out into other beakers such that all the beakers contain the same number of beans. If we need as many beakers as we have beans, the number of beans is Prime. So it is with 5. If we can have beakers with all the same numbers of beans & with more than 1 bean in each beaker, then the number of beans is Composite. For example with 28, we may have 4 beakers each with seven beans or seven beakers each with 4 beans, etcetera. See that the beakers are as variables and it makes no difference to the arrangements of the quantities what position the beakers have to one to another. At any point we may put all our beans in one beaker.

 

I do hope this has bean more helpful than knot. :)

[Don Blazys]

To: Modest,

 

When I wrote:

 

Originally Posted by Don Blazys

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!

 

all I meant to convey was that mathematics, in it's present form,

does absolutely nothing to preclude or prevent "indiscriminant" multiplication by unity,

which is, as we have already shown in this thread, "undefined".

 

Present day mathematics relies on us being polite, considerate and friendly enough

to express a simple number such as "seven", by writing [math]7[/math].

 

It (present day mathematics) does not take into account the possibility that

mathematicians who are impolite, inconsiderate and unfriendly can,

at any time, choose to annoy the rest of us by expressing "seven" as:

 

[math]((1*1)*((1*1)^7)*(1*1^7)*(((1*1*1*7)/(1^7))/1))[/math]

 

and then claim that it is "merely redundant", but not wrong, or impossible,

or inconsistent with number theory.

 

"Blazys terms", on the other hand, would never allow such shenanegans

because they simply don't allow "multiplication by unity".

 

Moreover, "Blazys terms" demonstrate that "multiplication by unity" is

inconsistent with the properties of logarithms and actually results in division by zero!

 

You see, when I wrote the above quote, I was merely pointing out the fact that

indescriminant multiplications by unity, when performed on numbers that are part of

some legitimate factorization, compromize the integrity of that factorization.

 

In other words, all I did was take the legitimate proper factorization of [math]6[/math]

and showed a particular one to one correspondence by writing:

 

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

 

Then, to point out the sheer stupidity of multiplication by unity,

I "indiscriminantly" multiplied the [math]1[/math] on the left by [math]1[/math] so as to obtain:

 

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

 

where the equation on the left now "appears" as a "factorization" but is,

in fact, nothing more than a vaccuous and therefore idiotic statement.

 

Then, to illustrate the utter absurdity of any attempt to now regain,

under this condition, a one to one correspondence,

I added [math]1[/math] to both sides of the equation on the right so as to show:

 

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

 

That's all there is to it!

 

I was simply illustrating how "redundant but harmless" multiplications by unity

don't belong in number theory.

 

None of this involved some "new way of factoring", or anything of that sort!

 

Don.

[Turtle]

all I meant to convey was that mathematics, in it's present form,

does absolutely nothing to preclude or prevent "indiscriminant" multiplication by unity,

which is, as we have already shown in this thread, "undefined".

 

Present day mathematics relies on us being polite, considerate and friendly enough

to express a simple number such as "seven", by writing [math]7[/math].

 

It (present day mathematics) does not take into account the possibility that

mathematicians who are impolite, inconsiderate and unfriendly can,

at any time, choose to annoy the rest of us by expressing "seven" as:

 

[math]((1*1)*((1*1)^7)*(1*1^7)*(((1*1*1*7)/(1^7))/1))[/math]

 

and then claim that it is "merely redundant", but not wrong, or impossible,

or inconsistent with number theory.

 

...

Don.

 

All those ones prompted me to recall a discussion of why 1 is not Prime. Four answers are given here, but it's #2 that appears most relevant to your expression here. Boldenation mine.

 

Why is the number one not a prime?

...Answer Two: Because of the purpose of primes.

The formal notion of primes was introduced by Euclid in his study of perfect numbers (in his "geometry" classic The Elements). Euclid needed to know when an integer n factored into a product of smaller integers (a nontrivially factorization)' date=' hence he was interested in those numbers which did not factor. Using the definition above he proved:

[u']The Fundamental Theorem of Arithmetic[/u]

Every positive integer greater than one can be written uniquely as a product of primes, with the prime factors in the product written in order of nondecreasing size.

Here we find the most important use of primes: they are the unique building blocks of the multiplicative group of integers. In discussion of warfare you often hear the phrase "divide and conquer." The same principle holds in mathematics. Many of the properties of an integer can be traced back to the properties of its prime divisors, allowing us to divide the problem (literally) into smaller problems. The number one is useless in this regard because a = 1*a = 1*1*a = ... That is, divisibility by one fails to provide us any information about a. ...

[Don Blazys]

To: Turtle,

 

Thanks for the info on Euclid and why unity is not a prime. When you wrote:

 

Quoting Turtle:

Many of the properties of an integer can be traced back to the properties of its prime divisors,

allowing us to divide the problem (literally) into smaller problems.

The number one is useless in this regard because a = 1*a = 1*1*a = ...

That is, divisibility by one fails to provide us any information about a. ...

 

you were not just "whistling Dixie"! :Whistle: :note2: :note:

 

Not only does "multiplication by unity" fail to provide us any information about

the properties of integers, it doesn't even provide us with any information as to

what operation is being performed!

 

"Multiplication by unity" is absolutely indistinguishable from "division by unity",

so both must be undefined.

 

[math]a*1=a/1[/math],

 

so multiplying and/or dividing by unity renders the

multiplication and division symbols utterly meaningless. Moreover,

 

[math]1*(a+b)=1*a+b=a+1*b=1*a+1*b[/math],

 

so there is no way to know if "multiplication and/or division by unity" is "distributive".

 

Multiplying and/or dividing by unity is like peddling a bicycle with a broken chain! It's a joke!

Sometimes I think that it has the one redeeming quality of being funny :lol: ,

but other than that, I can't think of even thing that it is good for. Now that "Blazys terms"

are making the rounds, how otherwise astute and talented mathematicians can continue

to subscribe to "multiplication and/or division by unity" is beyond me.

 

One of the really cool things about "Blazys terms" is that they actually demonstrate that

unity can be a multiplicand but not a multiplier, and this, in turn, allows us to define

"prime number" as:

 

"Any positive integer that can be both a multiplicand and a multiplier

whose only factors are unity and itself".

 

(Note that unity is excluded because it can't be both a multiplicand and a multiplier!)

 

Now, let's get back to "wordsmithing".

"Wordsmithing" is important!

Had mathematicians not found Euclids fifth postulate to be "too wordy"

and therefore "not worthy",

they would not have conducted further investigations ,

and would not have discovered "non-Euclidean geometry".

 

So, let us now contemplate the definition or "word equation":

 

multiplication=repeated addition.

 

Notice that without the word "repeated", we would simply have:

 

multiplication=addition,

 

and that, of course, would be just plain silly :D and therefore "unacceptable".

Clearly, the word "repeated", which means: "having occured more than once",

or, in mathematical language: "[math]>1[/math]" is crucial to the definition of multiplication,

so we must now investigate into how it applies in your

"bean and beaker" model.

 

Quoting Turtle:

Imagine now that you have an endless supply of glass beakers

in which to place arrangements of beans, which is to say amounts of beans as they will

self-arrange physically in the beaker. Imagine also an endless supply of beans.

 

Let's count out 5 beans and put them all in a single beaker.

This is multiplication by unity. One beaker of 5 beans.

Now we can proceed to take all but 1of those beans from the beaker and

set about seeing if we can put those beans taken out into other beakers

such that all the beakers contain the same number of beans.

If we need as many beakers as we have beans, the number of beans is Prime.

So it is with 5.

If we can have beakers with all the same numbers of beans

& with more than 1 bean in each beaker, then the number of beans is Composite.

For example with 28, we may have 4 beakers each with seven beans or

seven beakers each with 4 beans, etcetera. See that the beakers are as variables

and it makes no difference to the arrangements of the quantities what position

the beakers have to one to another. At any point we may put all our beans in one beaker.

 

Now, in order to introduce the all important element of "repetition" into this model,

let's equate "bean" to "multiplicand" and "beaker" to "multiplier".

Then, the multiplication [math]1*5[/math] can be expressed as either:

 

"[math]5[/math] beans evenly distributed in [math]5[/math] beakers",

 

which can be viewed as "multiplication of unity" where unity is the "multiplicand", or

 

"[math]5[/math] beans in [math]1[/math] beaker",

 

which, as you pointed out, can be viewed as "multiplication by unity", where unity is

the "multiplier".

 

Now, it is clear that in the first case, both beans and beakers are "repeated",

as there are [math]5[/math] of each, and [math]5>1[/math].

However, in the second case, the beaker is not "repeated" as there is only [math]1[/math].

Thus, unity can not be a multiplier (beaker) but only a multiplicand (bean).

Beans always repeat, but beakers don't.

 

Your "bean and beaker" model has made all this more clear than ever.

 

Thanks

 

Don.

[Turtle]

To: Turtle,

 

Now, let's get back to "wordsmithing".

"Wordsmithing" is important!

...

Now, it is clear that in the first case, both beans and beakers are "repeated",

as there are [math]5[/math] of each, and [math]5>1[/math].

However, in the second case, the beaker is not "repeated" as there is only [math]1[/math].

Thus, unity can not be a multiplier (beaker) but only a multiplicand (bean).

Beans always repeat, but beakers don't.

 

Your "bean and beaker" model has made all this more clear than ever.

 

Thanks

 

Don.

 

:eek: :hyper: On the boldened I have to ask, "what beaker?". Why the beaker I already mentioned, i.e. the 1 you repeated when you mentioned it. As soon as you refer to any "it", you have repeated "it". As I say though, I don't see what you're onto with the logrithms (when they mentioned it was upcoming in math class I thought we were going to drum on trees so I skipped. :hyper:), so I'll keep reading along for awhile. If by any chance you can use your Blazy constant to help me root out some Strange Anomolies, I'd be forever in your debt. :)

[modest]

It (present day mathematics) does not take into account the possibility that

mathematicians who are impolite, inconsiderate and unfriendly can,

at any time, choose to annoy the rest of us by expressing "seven" as:

[math]((1*1)*((1*1)^7)*(1*1^7)*(((1*1*1*7)/(1^7))/1))[/math]

 

But, if we’re allowed to use the normal axioms of the natural numbers where N•1=N, then the unwieldy thing above can be simplified.

 

Then, to point out the sheer stupidity of multiplication by unity,

I "indiscriminantly" multiplied the 1 on the left by 1 so as to obtain:

 

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

 

where the equation on the left now "appears" as a "factorization" but is,

in fact, nothing more than a vaccuous and therefore idiotic statement.

 

Ok, I think we’re on the same page. Multiplying a factor by unity does not imply adding one to its divisors.

 

Moreover, "Blazys terms" demonstrate that "multiplication by unity" is

inconsistent with the properties of logarithms and actually results in division by zero!

 

I’m not too sure what a Blazys term is, I joined this thread only recently. What I’ve seen is the identity:

[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]

To me, this doesn’t imply anything about multiplication by unity. As long as [math]T \neq 0[/math], both sides simplify algebraically to [math]a^x[/math]. Is this identity the source of “Blazys terms”?

 

Looking at the rhs of your identity,

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

And, for ease of readability, let’s look only at the exponent,

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

Substituting 1 = ln(t)/ln(t)

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

Subtracting fractions with like bases and rearranging,

[math]\frac{x\ln(a)-\ln(T)}{\ln(a)-\ln(T)}[/math]

substituting [math]x\ln(a) = \ln(a^x)[/math] using [math]\log_{b} (x^y) = y \log_{b}(x)[/math] (source),

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

using [math]\log_{b}(x)-\log_{b}(y) = \log_{b}(x/y)[/math],

[math]\cfrac{\left(\ln\Big(\cfrac{a^x}{T}\Big)\right)}{\left(\ln\Big(\cfrac{a}{T}\Big)\right)}[/math]

using [math]\log_{a}(b)=\frac{\log_{c}(b)}{\log_{c}(a)}[/math],

[math]\log_{a/T}\left(\frac{a^x}{T}\right)[/math]

This is log base (a/T) of (a^x/T). The right hand side of your identity is now,

[math]T\left(\frac{a}{T}\right)^{\left(\log_{a/T}\left(a^x/T\right)\right)}[/math]

using [math]b^{\log_{b}(x)}=x[/math],

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

The whole identity then,

[math]\frac{T}{T}a^x=T\frac{a^x}{T}[/math]

 

Where [math]T\neq0[/math] your identity simplifies to:

[math]a^x=a^x[/math]

 

Like I said before, I’m not a mathematician. I don’t know what implications this has (and I’d be very appreciative of any opinion offered on that), but I can’t imagine it holds any deep consequences against multiplication by unity. Unless (0/0) is considered unity, I don’t see how the above is applicable at all.

 

~modest

[Don Blazys]

To: Turtle

 

Quoting Turtle:

On the boldened I have to ask, "what beaker?".

Why the beaker I already mentioned, i.e. the 1 you repeated when you mentioned it.

As soon as you refer to any "it", you have repeated "it".

 

The really cool thing about your "bean and beaker" model is that

the definition:

 

multiplication = repeated addition

 

as it applies to unity can be demonstrated using physical objects.

We need not think of the word "repeated" as "mentioned" or "refered"

but simply as "more than one bean or beaker",

and as we will soon find out, "more than one sound"!

So, let's actually perform this demonstration as a "laboritory experiment",

using real beans and beakers! Got your beans and beakers ready? Okay, here we go!

 

Quoting Turtle:

Imagine now that you have an endless supply of

glass beakers in which to place arrangements of beans,

which is to say amounts of beans as they will

self-arrange physically in the beaker.

Imagine also an endless supply of beans.

 

Let's count out 5 beans and put them all in a single beaker.

This is multiplication by unity. One beaker of 5 beans.

 

Now, pick up each and every beaker, and shake it once.

All you will hear is "one sound" from the beaker containing the 5 beans.

The rest of the beakers will not produce a sound because they are empty.

Most importantly, note that the sound was not repeated. It occured only once.

 

Quoting Turtle:

Now we can proceed to take all but 1of those beans

from the beaker and set about seeing if we can put those beans taken out

into other beakers such that all the beakers contain the same number of beans.

 

This gives us 5 beakers containing one bean each.

Again, pick up each and every beaker, one at a time, and shake it once.

You will now hear 5 sounds from the five different beakers containing one bean each.

Most importantly, note that the sound was repeated. It occured five times.

 

Our "laboritory experiment" is now over.

We can now take our pens and clipboards and record our observations and results as follows:

 

First observation:

 

The sound that was not repeated occured when the "beakermultiplier" count was "[math]1[/math]"

and the "beanmultiplicand" count was "[math]5[/math] in a beaker".

Conclusion: If multiplication = repeated addition = repeated sound,

then [math]1[/math] is not a multiplier because [math]1[/math] "beakermultiplier" resulted in

a "sound" or "addition" that was not repeated.

 

Second observation:

 

The sound that was repeated occured when the "beakermultiplier" count was [math]5[/math],

and the "beanmultiplicand" count was "[math]1[/math] in each beaker".

Conclusion: If multiplication = repeated addition = repeated sound,

then [math]1[/math] is a multiplicand because [math]1[/math] "beanmultiplicand" in each beaker

resulted in a "sound" or "addition" that was repeated.

 

Quote Turtle:

As I say though, I don't see what you're onto with the logrithms

(when they mentioned it was upcoming in math class I thought

we were going to drum on trees so I skipped. ), so I'll keep reading along for awhile.

If by any chance you can use your Blazys constant to help me root out

some Strange Anomolies, I'd be forever in your debt.

 

We both share a deep interest in mathematical anomolies. Here's one:

The derivatives of "Blazys terms" can be modified to result in "prime counting functions"

that are more accurate than Li(x) to as far as Pi(x) has been calculated!

What makes them particularly special is that they are contained in either one or two terms!

Are you talking about stuff like that?

 

Don

[Don Blazys]

To: Modest,

 

Let's assume that all variables herein represent non-negative integers.

 

Quoting modest:

I’m not too sure what a Blazys term is, I joined this thread only recently. What I’ve seen is the identity:

 

[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]

 

The term:

 

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

 

is indeed a "Blazys term".

 

It's also known as a "cohesive term", and the properties that set it apart from

"non-cohesive" terms are in my "Proof of the Beal Conjecture" that you can find

on my website, (donblazys.com).

 

One of it's hallmarks is that intrinsically, it's variables are perfectly defined

by their own unique domains as:

 

[math]T= \{2, 3, 4...\}[/math], [math]a=\{1, 2, 3...\}[/math] and [math]x=\{0, 1, 2...\}[/math].

 

Compare this to the non-cohesive term:

 

[math]\frac{T}{T}a^x[/math]

 

whose variables are intrinsically defined by their domains as:

 

[math]T=\{1, 2, 3...\}[/math], [math]a=\{0, 1, 2...\}[/math] and [math]x=\{0, 1, 2...\}[/math].

 

Now, before we move on to how all this effects "multiplication by unity",

we must first answer a few questions. (Perhaps you or someone else will help me.)

The questions are as follows:

 

(1) Which term has the better defined variables?

 

(2) Is it logical to proceed from perfectly defined variables to poorly defined variables

and call it a "simplification"?

 

(3) Can the cancelled [math]T[/math]'s on both sides of the equation be "cancelled out"

meaning "crossed out" and made to "disappear"?

 

Don.