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Topic: Penn. State Univ Dr. Ecker reconsider's AP's No Odd Perfect Proof
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Registered: 3/31/08
Penn. State Univ Dr. Ecker reconsider's AP's No Odd Perfect Proof
Posted: Jan 26, 2014 3:21 PM
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On Tuesday, December 31, 2013 10:12:30 PM UTC-6, Dr. Mike Ecker wrote:
> I've been away for a while so I'll jump in, except I will use n instead of k.


Well, I changed the proof since Dr. Ecker first critiqued it. So here is the brand new proof, which is unsinkable as Mr. Bau would say--

Valid No Odd Perfect Proof , and predictions about Odd Abundant & Deficient Numbers

As I wrote earlier, the finest validation of a math proof is _not_ other fellow mathematicians opining it is true or being published in a math journal, but rather, the finest validation is whether the method of the proof itself goes on to enlighten further truths of that math topic. For instance, my Beal proof method instantly proves Fermat's Last Theorem FLT, and my Goldbach proof method instantly proves the Generalized Goldbach.

When mathematics has so called proofs that are only opined true or published in math journals like that of Appel & Haken's 4 Color Mapping or Wiles's FLT, for which their method has no more relation to doing anything more to mathematics, means that their offering is a fake work. Their offering is invalid.

So how am I sure that my No Odd Perfect proof is a true proof? Well, I take the method involved and see if it produces further truths about the topic in question such as odd deficient numbers and odd abundant numbers. My method says the proof is based on the fact that as 3 is the smallest factor of a odd number, then we have two groupings of the divisors, one grouping is 1/3 the number and the other grouping is 2/3 the number. So those groupings forbid a odd number to ever be perfect because of that 2 in the numerator of 2/3, since it means 2 is a divisor of the odd number. That is the mechanism of the proof.

So, now, if it is a valid proof, the mechanism or method should produce more facts about odd abundant and deficient numbers.

One fact is already clear, that the largest or maximum deficient number would miss being perfect by the amount of 2. So that means there must exist at least one odd number with a deficit of 2, and it is found in 9 where 1x9 and 3x3, where we have 1 + 3 + 3, which misses being odd perfect by only 2, and the 9 is the only nearest miss by 2 of all the odd numbers.

And the method of proof implies that the nearest miss for an abundant number to be odd perfect would miss by 2(3x5) which we have in 945.

Now I spent some time delineating the abundant numbers 945, 1575, 2205.







Now the sum of 945 is 975 with a abundance of 30 = 2(3x5).
The sum of 1575 is 1649 with a abundance of 74 = 2(37).
The sum of 2205 is 2241 with a abundance of 36 = 2(2x9).

Provided I did my arithmetic correctly, so we see a conformation with the proof method that the odd abundant numbers would all fail being odd perfect because of that 2 in the 2/3 grouping.

Now there is one more phenomenon I want to discuss now, is the sequence of odd abundant in that starting with 945, the sequence is just a adding of 630 to 945 to get the next such odd abundant number. And that would agree to the proof method for 630 is the 2/3 of 945. Now a important question arises as to whether there are any odd abundant numbers other than that sequence as listed here:

{945, 1575, 2205, 2835, 3465, 4095, 4725, 5355, 5775, 5985, 6435, 6615, 6825, 7245, 7425, 7875, 8085, 8415, 8505, 8925, 9135, 9555, 9765, 10395, 11025, 11655, 12285, 12705, 12915, 13545, 14175, ...} from

So, here is the question, are these the only odd abundant numbers or are there any odd abundants interspersed between that above sequence? In a sense the above is a validification of the proof method of 1/3 and 2/3 groupings. Because if 9 is the maximum that a odd deficient number gets close to being perfect and misses it by 2, and if 945 is the nearest miss to being odd perfect for the abundant odd numbers and misses by 30, then the proof method is truly a grouping of 1/3 and 2/3 and the 2 in the 2/3 forbids the construction of the odd perfect.

Now maybe that sequence list above is not inclusive of all the odd abundant numbers. Maybe it is a list of only those separated by 630. So I need to find out.
Constructive proof No Odd Perfect Number

The basic term used is _cofactors_, where a number has its cofactors paired.

Example is 6 and 15:

The number 6 has cofactors of 1 with 6, and, 2 with 3 and represented as this:

(1 + 6) + (2 + 3) = 12

The number 15 has cofactors of 1 with 15, and, 3 with 5 and represented as this:

(1 + 15) + (3 + 5) = 24

For 18 we have

(1 + 18) + (2 + 9) + (3 + 6) = 39

For 20 we have

(1 + 20) + (2 + 10) + (4 + 5) = 42

For 9 we have

(1 + 9) + (3 + 3) = 16

For 28 we have

(1 + 28) + (2 + 14) + (4 + 7) = 56

Also, let me focus on the number 945 since it is odd abundant so as to give the reader some bearings of odd abundant and odd deficient numbers.

(1 + 945) + (3 + 315) + (5 + 189) + (7 + 135) + (9 + 105) + (15 + 63) + (21 + 45) + (27 + 35) and once we omit the 945 the sum of divisors is 975.

I displayed this abundant odd number to compare with the deficient odd number of 15. Few people know that some odd numbers can be abundant. Why is that important? Because if the odd numbers can overshoot and undershoot the mark, stands to reason that perhaps some odd number falls smack on the spot of equal.

Constructive Definition of a Perfect Number
Now let me define the Perfect Number in general as that of omitting the number itself k as a divisor, the remaining cofactor divisors add up to k.

For example, 6 and omitting 6 has 1+2+3 =6. And 28 omitting 28 has 1+2+4+7+14.

Construction proof

Take the arbitrary Odd Perfect Odd number larger than 1 and call it k.

We ask just one simple question for the proof. We ask, what is the smallest divisor for a odd perfect number? It cannot be 2 for that means k is divisible by 2 and no longer odd. That means the smallest divisor is 3 or 5 or 7 etc etc. The proof that 3 cannot be the smallest divisor case takes care of the 5,7,9 etc cases. So I no longer will talk about if 5 etc is the smallest divisor.

Let us construct the arbitrary odd perfect number that has its smallest divisor of 3. This means we can group all the divisors into just two groups of 1/3k and 2/3k.

Now since this odd perfect is grouped into 1/3 and 2/3, means that it has a possibility of these and only these permutations since k is odd:

1/3 2/3
3/9 6/9
5/15 10/15
7/21 14/21
315/945 630/945

So, now, can we construct this Odd Perfect Number given that definition?

The answer is no, because in all possible permutations of an odd number with a 1/3 to 2/3 grouping of added terms, we can never get rid of the even number 2 in that 2/3. This means that k is divisible by an even number 2 in order for the Odd Perfect to sum to k.

Let me illustrate that with another picture.
The grouping 1/3 and grouping 2/3 becomes this:

1/3 grouping + 1/3 grouping + 1/3 grouping

So for 15 to be Odd Perfect we would have this:

5 + 5 + 5

Which means 15 is divisible by 2 because 5 is the 1/3 grouping and the 2/3 grouping is 2x5 where 2 divides into 15.

So for 45 to be Odd Perfect we would have this:

15 + 15 + 15

Which means that 45 is divisible by 15 but it has another grouping of 2/3 composed of a 15x2 where 2 divides into 45.

So we see here how the construction of the Odd Perfect Number has an imposed barrier to construction, in that we can never get rid of the 2 in the 2/3 grouping and which forces the odd perfect k to be divisible by 2.

Now this proof must show that even perfects are possible and have no barrier to having some even numbers be perfect. That is "some even numbers" can be perfect.

Are the Even Perfects constructible by this proof method? Well, the even perfects are 1/2 and 1/2 so we have this:
1/2 1/2
2/4 2/4
3/6 3/6
14/28 14/28

And we see there is no barrier to force the groupings of 1/2 with 1/2 or 3/6 with 3/6 where the numerator cannot divide into the denominator. All of the 1/2 to 1/2 groupings allow the numerator to divide into the denominator, so there is no barrier to construction. However, in the case of odd perfect the 2/3 grouping never allows 2 to divide into 3 and is that barrier. So for odd numbers, the barrier to construction of odd perfect is the perpetual even number 2 divisor in the groupings of 1/3 and 2/3 and its permutations.


Proof that Perfect Numbers is a finite set

Now in mathematics the two oldest unsolved problems deal with the Perfect Numbers, the No Odd Perfect (except 1) conjecture and the question of whether perfect numbers are finite or infinite. I proved the No Odd Perfect Numbers here in sci.math in the last several weeks. I proved the Finitude of Perfect Numbers years ago, but let me repeat it here for the history record.

Proof of the Finitude of Perfect Numbers

When mathematics is honest about its definitions of finite versus infinite, it seeks a borderline between the two concepts, otherwise they are just one concept. From several proofs of regular polyhedra and of the tractrix versus circle we find the borderline to be 1*10^603. That causes a measure to use for all questions of sets as to whether they are finite or infinite. The Naturals are infinite because there are exactly 1*10^603 inside of 1*10^603 (not counting 0).
The algebraic-closure of numbers is 1*10^1206 which forms a density measure. So are the primes finite or infinite? Well, are there 1*10^603 primes between 0 and 1*10^1206? Easily for at around 10^607 we have 10^603 primes. Are the Perfect-Squares {1, 4, 9, 16, 25, . .} finite or infinite set? The perfect-squares are a special set since they are "minimal infinite" since there are exactly 10^603 of them from 0 to 10^1206.

How about perfect cubes? Well they are a finite set since there is not enough of them between 0 and 10^1206.

How about Fibonacci primes or Mersenne primes? Both of them are so rare, that there are only a handful between 0 and 10^1206.

So this is the modern means of checking whether a set is finite or infinite. We ask the density of the set from 0 to 10^1206 and if there are 10^603 of the objects in that space, then they form an infinite set. If not, they are finite.

Looking at the density of Perfect-Numbers and the list is { 6, 28, 496, 8128, . . .} and we immediately see the density is nowhere close to the Minimal density of Perfect-Squares {1, 4, 9, 16, 25, . . .} and so we immediately can see the Perfect-Numbers forms a finite set.


Recently I re-opened the old newsgroup of 1990s and there one can read my recent posts without the hassle of mockers and hatemongers.!forum/plutonium-atom-universe        

Archimedes Plutonium

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