Perfect Numbers: An Aural Dissertation

[CraigD]

Zero as a single digit represents for me at least a blank and is not a number. …

It’s important to understand the distinction between a numeral and a number.

 

A numeral is any collection of symbol that represents a number. “12310”, “3247”, “11111012”, “CXXIII” and “one-hundred twenty-three” are all numerals representing the same number.

 

The “zero character” is a symbol in some, but not all, numeral systems (eg: the binary and decimal numeral systems, but not the Roman numeral system). “Zero” is a number in some, but not all number systems (eg: The integers and rational numbers, but not the natural numbers). In the same way that the name for a thing is not the same as the thing, the symbols representing a number are not the same thing as the number. The failure to make this distinction is a common kind of category error.

[Turtle]

___Now this, is aural! I intended to leave things simmering over the holidays, but I see the pot needs stirring.

___Welcome Craig, Qfwfq, JayQ, Racoon, et al. & thanks for posting. I have read & understood your replies, & offer the following observations.

___The definition of Perfect Number is most often related to the Pythagorians & required no "numerals" other than words. That is to say, it is possible to discuss this "orally" (therfore my play on words in the title, "aural" for "oral"). Try reading the thread aloud to get the feel of things. To whit: A Perfect Number is one which is equal to the sum of its parts. "Number means "Cardinal Number" as per the definition of "one" above. "Parts" means even divisors, & sum means to add them. One finds Perferct Numbers by factoring, much as Erastosenes (sp) sieved for Primes. The algorithm is as follows: Take a cardinal number (zero isn't a cardinal number so it is not any consideration in this), & find the proper/even divisors. All cardinal numbers divide evenly by both one & themselves, but we don't count the number itself in the sum. Now add one, plus each of any other even divisors & compare that sum to the original number. If the sum is less than the original number, the number is termed "Deficient"; if the sum equals the number it is termed "Perfect"; if the sum excedes the number, the number is termed "Abundant".

___I have to yield the machine, but I won't leave things to hang any longer than necessary. One note on Craig"s definition. The only Perfect Numbers which have the product of divisors equal to them are one & six. Consider that twenty eight is a Perfect Number which divides evenly by one, two, four, seven, and fouteen. They sum to twenty-eight, but the product is seven-hundred-eighty-four.

____Gotta run.:cup:

[Turtle]

___Jumping back in, we continue. Perfect Numbers comprise a set in the same manner as do Prime Numbers. I took such great care in defining "one" at the start because six is universally regarded as the "first" Perfect Number. Under the definition of "one" & "perfect Number", I assert that one (1) is the first Perfect Number & that the assertion follows logically from these definitions.

___One, by definition, starts all counting. If you factor one, you derive a multiplier of one, a multiplicand of one, & a product of one. It divides evenly in other words, by one & itself. Not counting itself in the sum, we have one left & that is the sum which is equal to the natural number we started with, ie. one. As the sum equals the number, one is therefore a Perfect Number & moreover it is the First Perfect Number in the Set of Perfect Numbers.

___As it is "counting" we do, go get some beans or other small numerous "ones" & find the even divisors of a count by making little piles. For example, with six beans it is possible to evenly make either one pile of six, or six piles of one, or two piles of three, or three piles of two. Nothing more, nothing less. Try this with one bean, two beans, three beans etc..

___Time to simmer down now.:cup:

[Turtle]

It’s important to understand the distinction between a numeral and a number.

 

A numeral is any collection of symbol that represents a number. “12310”, “3247”, “11111012”, “CXXIII” and “one-hundred twenty-three” are all numerals representing the same number.

The “zero character” is a symbol in some, but not all, numeral systems (eg: the binary and decimal numeral systems, but not the Roman numeral system). “Zero” is a number in some, but not all number systems (eg: The integers and rational numbers, but not the natural numbers). In the same way that the name for a thing is not the same as the thing, the symbols representing a number are not the same thing as the number. The failure to make this distinction is a common kind of category error.

 

___Craig points out here - & quite nicely - that writing a "number" longhand IS a numeral. Thirty-six, for example is a numeral in the strictest sense of the word. As we have ventured into deep waters here, strictness & rigor must receive our every attention.

___As we have developed this topic I have adopted the convention of using the "longhand" numerals as a device to ameliorate our bias to positional based numeration systems. Craig has contributed numerous times to this very point with computer generated lists of sets in succesive bases, & as we progress here I expect to find similar utility in this approach. Seems every time Craig produces such a list, the mystery at hand only deepens.

___On an operational note, I early on asserted my intention to prove all Perfect Numbers evenly divide by two, a Strong Conjecture which remains unproven. I do not have that proof in my pocket, or indeed know its exact character or location, but I believe such a proof is possible & hope to come by it by way of our discussion.

___Let's count on one-another to root this out.:cup:

[Turtle]

___Find below the first seven elements of the set of Perfect Numbers expressed in base ten notation (including one[1] as the first set element per my argument above).

 

I have separated set elements with colons.

 

Set of Perfect Numbers {1: 6: 28: 496: 8,128: 33,550,336: 8,589,869,056...}

 

:cup:

[Turtle]

___As the proper divisors have the floor so to speak in regard to our Perfect Numbers, it seems reasonable to deliniate them. Here follow the first six Perfect Numbers & their associate proper divisors.

 

1---(1:1)

6---(1:6) (2:3)

28---(1:28) (2:14) (4:7)

496---(1:496) (2:248) (4:124) (8:62) (16:31)

8128---(1:8128) (2:4064) (4:2032) (8:1016) (16:508) (32:254) (64:127)

33550336---(1:33550336) (2:16775168) (4:8387584) (8:4193792)

(16:2096896) (32:1048448) (64:524224) (128:262112) (256:131056)

(512:65528) (1024:32764) (2048:16382) (4096:8191)

 

:)

[Turtle]

___Now no mention of Perfect Numbers is really complete until it includes the latest Largest Perfect Number, and it runs to 1,819,050 digits. It is of course evenenly divisable by 2 & this is readily seen as the number ends in 6 in base ten notation.

___The following link has the complete decimal extraction; all of the one-million-eighthundred-nineteen-thousand-fifty base ten digits of it!

 

http://calendarhome.com/prime/perfect4.html

 

Enjoy! :xparty: (By the way, it has a Katabatak transform of 1, as do all Perfect Numbers but 6; in other words, except for 6, all Perfect Numbers mod 9 = 1):eek:

 

 

 

 

:lol:

[Turtle]

___The first five Perfect Numbers in Binary.

 

 

1

110

11100

111110000

1111111000000

 

:xparty:

[Turtle]

___The first five Perfect Numbers in,

Binary - Base Two

1

110

11100

111110000

1111111000000

 

:)

 

Trinary - Base Three

1

20

1001

200101

102011001

 

Base Four

1

12

130

13300

1333000

 

;)

[Turtle]

Mars: "So Turtle; I hear you're going around saying 1 is the first perfect number".

 

Turtle: "Hi Mars; yes I asserted that. How does it sit with you?"

 

Mars: "I think it will cost too much to change all those text books asserting that 6 is first".

[Turtle]

Now forgetting 1 as Perfect for the moment, what the heck have I found here?

For N a 6-sided number where n is the ordinal, the expression generating the ordered set is N= (n/2)*(4n-2). [2n^2-n]

 

n=2:N=6

n=4:N=28

n=16:N=496

n=64:N=8,128

n=?:N=33,550,336 ?

n=? :N= 8,589,869,056 ?

 

http://hypography.com/forums/physics-mathematics/1343-katabatak-math-exploration-pure-number-theory-27.html

posts #261 & #262

[Turtle]

2n^2-n = 33,550,336

Solve for n where n is an integer

 

Ok...I'm looking at this for 2 hours now & drawing a complete blank as to how to go about solving it. (other than trial & error that is.;) )

Little help here?:wink:

[Turtle]

2n^2-n = 33,550,336

Solve for n where n is an integer

 

Ok...I'm looking at this for 2 hours now & drawing a complete blank as to how to go about solving it. (other than trial & error that is.;) )

Little help here?:cup:

 

It's looking to me like we have some diophantine equations on our hands here.:)

http://en.wikipedia.org/wiki/Diophantine_equation

[ughaibu]

I've had this bookmarked but haven't tried reading it, so apologies if it's no fun: http://216.239.51.104/search?q=cache:I2tEQLnJ1LMJ:math.ucsd.edu/programs/undergraduate/0405_honors_presentations/jpearlman_odd_perfectnumbers.pdf+%22odd+perfect+numbers%22+proof&hl=ja&gl=jp&ct=clnk&cd=18

[Turtle]

I've had this bookmarked but haven't tried reading it, so apologies if it's no fun: http://216.239.51.104/search?q=cache:I2tEQLnJ1LMJ:math.ucsd.edu/programs/undergraduate/0405_honors_presentations/jpearlman_odd_perfectnumbers.pdf+%22odd+perfect+numbers%22+proof&hl=ja&gl=jp&ct=clnk&cd=18

 

Excellent piece & right on topic ughaibu. Of primary note is that the paper deals with special cases only of a theoretical premise of the existence of odd perfect numbers.

Here what I have just discovered is a general expression that demonstrates the real situation that all known Perfect Numbers are even, and furthermore by this expression I have conjectured that all Perfect Numbers are also hexagonal, that is 6-sided numbers.

We have now to support my conjecture by solving the specific diophantine equation 2n^2-n = 33,550,336.:)

[Turtle]

...Here what I have just discovered is a general expression that demonstrates the real situation that all known Perfect Numbers are even, and furthermore by this expression I have conjectured that all Perfect Numbers are also hexagonal, that is 6-sided numbers.

We have now to support my conjecture by solving the specific diophantine equation 2n^2-n = 33,550,336.:cup:

 

I haven't teased out the solution yet for the above equation, but I intend to by at least trial & error this evening.

In the mean time I have rooted out 2 more Perfect Numbers for application to the same process. Here's the earlier list with the two new additions; I note in passing that I deliberately dis-included one as the first Perfect, but it does satisfy the equation 2n^2-n.

[n=1:N=1]

n=2:N=6

n=4:N=28

n=16:N=496

n=64:N=8,128

n=?:N=33,550,336 ?

n=? :N= 8,589,869,056 ?

n= ?:N=137,438,691,328?

n= ?:N=2,305,843,008,139,952,128?

:) ;)

[Turtle]

I haven't teased out the solution yet for the above equation, but I intend to by at least trial & error this evening.

In the mean time I have rooted out 2 more Perfect Numbers for application to the same process. Here's the earlier list with the two new additions; I note in passing that I deliberately dis-included one as the first Perfect, but it does satisfy the equation 2n^2-n.

[n=1:N=1]

n=2:N=6

n=4:N=28

n=16:N=496

n=64:N=8,128

n=4096:N=33,550,336

n=? :N= 8,589,869,056 ?

n= ?:N=137,438,691,328?

n= ?:N=2,305,843,008,139,952,128?

:) :cup:

 

Found it! The fifth Perfect Number is the 4096th hexagonal number. for n=4096, 2n^2-n = 33,550,336. Looks like a nice pattern of solutions where n is an even power of 2; not your Mersenne approach I dare say. On to the next: a solution for n where 2n^2-n=8,589,869,056

 

;)