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Topic: professors of Cambridge Univ endorse proof of No Odd Perfect to arxiv
Replies: 20   Last Post: Aug 25, 2014 1:52 AM

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 plutonium.archimedes@gmail.com Posts: 18,572 Registered: 3/31/08
professors of Cambridge Univ endorse proof of No Odd Perfect to arxiv
Posted: May 19, 2014 1:36 AM

Now No Odd Perfect and the Finiteness of Perfect Numbers are the oldest unsolved math problems in the books. Cambridge Univ is the oldest school I know of, so it is appropriate to ask Cambridge math professors to endorse these two proofs to Cornell's arXiv.org.

Detailed Proof that No Odd Perfect exists and proof that Perfect Numbers form a Finite Set

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. I simultaneously take a second grouping of the number odd Y as 1/Y and even/Y. Using both groupings I discover that an even number is a divisor of an odd number Y. So those groupings forbid a odd number to ever be perfect because of that 2 in the numerator of 2/3 or the even/Y, 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 **nearest miss** of a 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 turns out there are two such numbers, the number 3, and 9 where 1x9 and 3x3, where we have 1 + 3 + 3, which misses being odd perfect by only 2, and the 3, and 9 are 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.

945
___

3x315
5x189
7x135
9x105
15x63
21x45
27x35

1575
____

3x525
5x315
7x225
9x175
15x105
21x75
25x63
35x45

2205
____

3x735
5x441
7x315
9x245
15x147
21x105
35x63
45x49

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 http://oeis.org/wiki/Odd_abundant_numbers

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 3 and 9 are the closest that a odd deficient number gets 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 are 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 other than 1? 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 will prove in the same manner that 5,7,9 etc cases cannot be the smallest divisor. So I no longer will talk about if 5 etc is the smallest divisor.

Let us construct the arbitrary odd perfect number k, 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 where the k is the odd perfect number.

Now since this odd perfect k is grouped into 1/3k and 2/3k, 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
15/45 30/45
.
.
.
315/945 630/945

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

Well, we need another construction first. We need the construction of two more groupings of the Odd Perfect Number k as that of 1/k with its even/k where 1/k group plus the sum of even/k group equals k itself.

For example the odd number 15 has a 1/3 group plus 2/3 group as that of 5/15 + 10/15, and has the groupings of 1/15 + 14/15.

All Odd Perfect Numbers have these two different and separate groupings:

(1) Smallest odd divisor not 1 as 1/3 group + 2/3 group
(2) 1/k where we do group k into 1/k and even/k

Now we realize the proof that no such construction of a Odd Perfect can exist because both different and separate groupings have a even number in the numerator of both separate groupings that forces a 2 to be a divisor of the Odd Perfect number k.

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

Let me illustrate that with the number 15 and why it cannot be Odd Perfect.
The grouping 1/3 and grouping 2/3 becomes this:

5/15 grouping + 10/15 grouping where 5 + 10 = 15. Meaning that there are divisors in 15, at least two odd divisors that sum to 10.

Now let us take the simultaneous other groupings based on 1 as divisor:

1/15 grouping + 14/15 grouping where 1 + 14 = 15. Meaning that there are divisors of 14 such that at least two odd divisors that sum to 14.

Here we get an impossible construction of requiring at least two odd divisors sum to both 10 and sum to both 14.

The only possible solution is that 15 is divisible by 2 and so we would have 8 +2 = 10 and 8 + 6 = 14 with 2 a divisor of 15.

Now I spoke earlier that the proof for the cases when the smallest divisor is 5, not 3 follows the same method, or the proof that the smallest divisor is 7 (not 3 or 5) follows the same method.

The method is Two Separate Groupings of the Odd Perfect k, where one grouping is the smallest divisor of 3 or larger, and the second grouping is based on 1 being the smallest divisor. When you combine the two separate groupings and inspect what possible divisors can deliver this Odd Perfect Number, you quickly realize the only divisors to simultaneously solve both separate groupings is the even number 2 and all its even composites.

QED

________________________________________
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. I proved the Finitude of Perfect Numbers years ago, but let me repeat it here for the history record.

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 area we find the borderline to be 1*10^603. That causes a measure gauge rod 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, since from the Prime Counting Function x/Ln(x) 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. I know of no other minimal infinite set different from Perfect Squares.

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.