A Mathematical Emergency.

Turtle · 01-25-2009

Quote:

Originally Posted by modest

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

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.

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.

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

Quote:

Originally Posted by Craig & Turtle

Quote:

Originally Posted by Turtle

[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.

CraigD · 01-25-2009

Quote:

Originally Posted by Don Blazys

and $28$ is called a "perfect number" because:

$1+2+4+7+14=28=(1*2*4*7*14)^{\left(\frac{1}{2}\right)}$ .

Note that the proper factors of $6$ "multiply perfectly" as a "first root",
while the proper factors of $28$ "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, $496$ , 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".)

Quote:

Originally Posted by modest

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

I think Modest is correct.

Also, the perfect numbers are given the formula $2^{n-1}(2^n -1)$ , where $n$ 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, $M_n = 2^n -1$ . 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 $n$ . For example, the product of the proper factors of the 10th perfect number,
$191561942608236107294793378084303638130997321548169216$
is
$191561942608236107294793378084303638130997321548169216^{88}$

modest · 01-25-2009

Quote:

Originally Posted by Turtle

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.

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/ugre...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.

Quote:

Originally Posted by CraigD

Also, the perfect numbers are given the formula $2^{n-1}(2^n -1)$ , where $n$ 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, $M_n = 2^n -1$ . 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 $n$ . For example, the product of the proper factors of the 10th perfect number,
$191561942608236107294793378084303638130997321548169216$
is
$191561942608236107294793378084303638130997321548169216^{88}$

Not enough +rep to go around

~modest

Don Blazys · 01-26-2009

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!

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 · 01-27-2009

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:

Quote:

Originally Posted by modest

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, $\sigma(n)$ . I will now give examples to clarify:

The number 12 is not perfect because,

$s(12) = 1+2+3+4+6 = 16$

$s(12) \neq 12$

The number 28 is perfect because,

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

$s(28) = 28$

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 · 01-27-2009

Quote:

Originally Posted by Don Blazys

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:

Code:

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 · 01-28-2009

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:

Quote:

One is a factor of 28 once.

Quote:

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 $1$ is a factor of $28$ only once.
But, here is the real question:

If we have the multiplication:

$1*28$ ,

then did we begin with $1$ and increase it by a factor of $28$ ,

or did we begin with $28$ and "increase" it by a factor of $1$ ?

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":

$1*1$

can only be interpreted as:

"We began with $1$ , then "increased" it by a factor of $1$ ".

As I mentioned in post #97, in number theory, multiplication is strictly and stringently
defined as "repeated addition", so that the multiplication: $3*5$ is viewed as either:

$3+3+3+3+3$ or $5+5+5$ ,

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: $1*28$ , then we get either:

$1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1$ or $28$ ,

Notice that the representation:

$28$

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: $1*1$ yields:

$1$ ,

and it is now painfully obvious that the representation: $1$
does not "repeat", and that the representation: $1*1$
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 · 01-28-2009

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 · 01-28-2009

Quote:

Originally Posted by Don Blazys

...
But, here is the real question:

If we have the multiplication:

$1*28$ ,

then did we begin with $1$ and increase it by a factor of $28$ ,

or did we begin with $28$ and "increase" it by a factor of $1$ ?

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 · 01-29-2009

Quote:

Originally Posted by 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.

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:

Quote:

Originally Posted by Don Blazys

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!

It reads to me like you are saying that if we are allowed to factor 28 as (1 • 1 • 2 • 2 • 7) or $(1^2 \cdot 2^2 \cdot 7) = 28$ rather than the normal factorization $(2^2 \cdot 7) = 28$ 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:

$(2^2 \cdot 7) = 28$

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

$D = (a^x \cdot b^y)$

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:

$2^0 \cdot 7^0 = 1$
$2^0 \cdot 7^1 = 7$
$2^1 \cdot 7^0 = 2$
$2^1 \cdot 7^1 = 14$
$2^2 \cdot 7^0 = 4$
$2^2 \cdot 7^1 = 28$

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,

$(1^2 \cdot 2^2 \cdot 7) = 28$

to the divisors of 28, I would say,

$D = (a^x \cdot b^y \cdot c^z)$

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:

$1^0 \cdot 2^0 \cdot 7^0 = 1$
$1^0 \cdot 2^0 \cdot 7^1 = 7$
$1^0 \cdot 2^1 \cdot 7^0 = 2$
$1^0 \cdot 2^1 \cdot 7^1 = 14$
$1^0 \cdot 2^2 \cdot 7^0 = 4$
$1^0 \cdot 2^2 \cdot 7^1 = 28$
$1^1 \cdot 2^0 \cdot 7^0 = 1$
$1^1 \cdot 2^0 \cdot 7^1 = 7$
$1^1 \cdot 2^1 \cdot 7^0 = 2$
$1^1 \cdot 2^1 \cdot 7^1 = 14$
$1^1 \cdot 2^2 \cdot 7^0 = 4$
$1^1 \cdot 2^2 \cdot 7^1 = 28$
$1^2 \cdot 2^0 \cdot 7^0 = 1$
$1^2 \cdot 2^0 \cdot 7^1 = 7$
$1^2 \cdot 2^1 \cdot 7^0 = 2$
$1^2 \cdot 2^1 \cdot 7^1 = 14$
$1^2 \cdot 2^2 \cdot 7^0 = 4$
$1^2 \cdot 2^2 \cdot 7^1 = 28$