Strange Numbers

[Turtle]

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Edited September 21, 2013 by Turtle

[Tormod]

Sorry; I gave the wrong link:

http://home.comcast.net/~turtlediable/wsb/index.html

Hey Turtle - do you mind explaining to my non-mathematical mind what this is all about? I like number theory but didn't instantly get what these numbers are meant to represent.

[maddog]

Huh ?

 

Why is 24 strange. Why is it's sum(?) (to what ? 12?) strange. I'm missing it. :hihi:

 

Maddog

[Bo]

i also don't see more then a way to generate a set of numbers... What's special?

 

 

Bo

[maddog]

Begin with the idea of perfect numbers;OK? Now if a number's divisor sum equals that number it's perfect, if the sum is less it's called deficient, & if the sum is more it's called abundant. So all Strange numbers are abundant by exactly 12(twice the first perfect number six) The set of Bizarre numbers abundant by exactly 56 (twice the second number 28). The sets' discovery I believe is original to me; I have never seen them described. They are further special because most of the members of each set have the same number of divisors. Because some don't fit the general pattern of set elements having a perfect number as a divsor (304 for example in the Strange set), the only way to find these sets is to factor the integers sequentially. Since factoring is considered a "hard problem" in computer science, looking for large elements to the sets as well as elements that don't fit the general rule takes weeks.

It's interesting because no one ever found them before & I did & this topic has been studied for thousands of years.

Lay it on me.

 

I see I had forgot the definition of Perfect numbers, Thank You. So why does the Definition

of Strange numbers need to be abundant by 12 ??? So to hear you out..

 

The divisors of 12 are 1, 2, 3, 4, 6 in which the sum is 16 and thus abundant.

The divisors of 24 are 1, 2, 3, 4, 6, 12, in which the sum of 28 and thus abundant.

 

So this makes 24 and 36 strange because they are both 12 more than an abundant

number ? Am I getting this right ? :hihi:

 

Maddog

[maddog]

I would conjecture from your statement that every multiple of 12 beyond 12 is a strange

number using you definition.

 

To check the next few

 

36: 1, 2, 3, 4, 6, 9, 12, 18 => sum = 55 > 36 => abundant

48: 1, 2, 3, 4, 6, 8, 12, 16, 24 => sum = 76 > 48 => abundant

60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 => 108 > 60 => abundant

 

So what does this mean ??? :hihi:

 

Maddog

[maddog]

Ouch; just lost a twenty minute respose into the ether! Rats.

Ok 60 is abundant but not Strange because 108-60=48. To be Strange the difference must be 12. You can find most Strange numbers by multiplying 6 (a perfect number) times a prime. But this does not find 304, which does not divide by 6(or 12). Also most Strange numbers have exactly 8 divisors (4 pairs) & 304 has 10 (5 pairs) This why it's strange. 304 I call an anomalous Strange Number & the next one is 127,744. The other sets also have these anomolys & so the only way to find them all is factoring.

What does it mean? It means I found something everyone for a couple thousand years has overlooked. It's like if you found a new cave, you would explore on your own as much as possible & then begin showing people what you found. Let them decide what it means; but you always remain the ONE who found it first.

 

OK... :rant: So, why does difference "need" to be 12 ? (2 * 6 -- a perfect#)... ??? Couldn't

it be any other value ? What is so special about 12 ???? :hihi:

 

Maddog

[maddog]

It is not that the difference "needs" to be 12, & yes, it can be ( and is) almost everything else. The simple point is, all those other differences that "be", don't form up any consistant sets. Differnces of twice a perfect do. I don't know why, I only know the property exists. Keep at me until you understand.

 

I read your explanation and I now getting it. I am wondering if we are seeing hear something

like a number theory version of fractals. I see playing with the two operations of addition and

multiplication over the integers, thus we are dealing with ring. So I am wondering if these

form some prime ideals. I am sorry if I am sound gibberish; these are terms from Group

Theory and Abstract Algebra. I will ask a friend of mine who is a Number Theorist here and

see what he thinks. I am intrigued.

 

What if I were to construct this over the complex integers {x + iy} and do the same amount

of computation over its modulus ? I probably won't get a chance to discuss with him until

next week. Good Luck!

[maddog]

Turtle,

 

I talked with my Number Theorist friend and he said he had played with some of this earlier

in his development. He showed me a generalization by picking another example based on

28 (another Perfect number) as follows:

 

Let p be a prime number, q be a perfect number, g and k be an integer. Any number of the

form

 

g = q * p^k

 

will also have Strange properties as you mention. They need not nessecarily be always 12

more (but any perfect number)

 

This does increase the overall number of Strange numbers and change their distribution.

 

A corrolary I am now wondering, are all Perfect numbers divisible by 2 ? ;)

 

Maddog

[maddog]

Now we're getting somewhere. Your friend's general expression is not enough & too much. You don't need to raise the prime to any power at all & it will not produce all set memebers.

 

My memory is not too good. I must have added the powers. :( BTW, he was calling them

Wierd numbers with the same definition as you describe.

 

304 is a Strange Number & has no perfect factors. Now I call the set Strange when the divisor sum is 12 more, Bizarre set numbers divisor sum 56 more, peculiar 992 more, Curious set is 16,256 more; yes each excedes by 2*perfect. Of course the question of proving if all perfect numbers are even is an unsolved problem in number theory; a kind of holy grail still. Now I don't accept that your friend is at all describing what I found & still claim I am the first to describe these sets. Search as you will, you will find no list elewhere which arranges these numbers in a set as I have, nor any expression that produces them. ;)

 

Turtle - how is 304 Strange when it is the product of 2^4 * 19 = 304. So the factors:

 

304: 2, 4, 8, 16, 19, 38, 76, 152 => sum = 315 which is 11 more than 304 and Not 12 ? ;)

 

I don't know if he found what specifically you found (?), what my friend said is a way to

represent "all" the numbers as

 

g = q * p where q is a perfect integer and p is a prime (sorry for the powers earlier). ;)

 

Maddog

[IrishEyes]

if you guys keep this up, we're going to have to add a MATH forum... ;)

seriously though, this stuff is awesome. you guys just blow me away!

[maddog]

Irish,

 

I have begun to think of this thread moving to the Physics/Math forum. ;)

 

Turtle,

 

I told my friend about 304 bieng the product of 2^4 and 19. He did think that wierd. From

him restating the definition of Wierd numbers and the lowest one being 70, I see these are

definitely different kind of numbers.

 

In the meantime, I found some websites on a related subject of Mersenne primes and perfect

numbers.

 

http://mathworld.wolfram.com/PerfectNumber.html

http://www.utm.edu/research/primes/mersenne/

http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Perfect_numbers.html

http://mathforum.org/dr.math/faq/faq.perfect.html

 

I have thought about it and figure that any normed division ring where the modulus was

used to calculate as you have wouldn't work directly. However, there is curiousity based

on one of the above websites ==> what would the numbers you found look like in other

bases ? ;)

 

Maddog

[maddog]

Yes, I think moving to Math section is good; I didn't know how to do it. I believe I was visiting just those links you listed when I found Hypography! As I have worked a lot in other bases & have written software to operate in base 2 to 29, in which base of this range would you like to see the list of Strange Numbers? Now I do use the convention for bases over ten of J=10, K=11 etc & I am discussing this in another thread. Nonetheless, the approach you suggest Madddog is very good. I don't like to type long posts, but I think this moves us forward.

 

The convention for any bases higher that 10 is to use A thru Z for bases 11 to 36. For

higher you could upper/lower case... for base 62.

 

1 .. 9, A .. Z, a .. z : be 1 .. 61 for base 62.

 

I would think interesting bases would be of prime, perfect or strange order. Thus of

interest would be

 

bases: 6, 7, 11, 12, 13, 17, 19, 24, 28, 36, 56, for starters. Some of these would be of

interest to me directly in physics, namely: 6, 7, 11, 12, 13, 24, 28, 56.

 

I don't know what pattern may form. I just feel it might be interesting. ;)

 

Maddog

[IrishEyes]

Thread moved to Physics/Math Forum

 

Keep crunching those numbers...

[maddog]

Thanks Irish. ;) So before moving on to another topic, do I understand maddog that you & your friend now believe I found something previously undescribed? I just want to be sure I'm advancing from a solid base. (Hahahaha; that's math humor!) :(

 

Enough that you have intrigued him and that he has heard of this and he was at the last Forum on Number

Theory just three weeks ago in Las Vegas!

 

BTW, he and the website I mentioned has stated the holy grail (oldest known) of conjectures that (drum

roll please)...

 

All perfect numbers are even... Ta Da!!! ;)

 

This one has been in search of an answer either way since Pythagorus' time. By way of example of an

Odd Perfect Number or that one Definitely DOESN'T exist. Were you to prove/disprove that, you could be

famous! :(

 

Maddog

[alxian]

ok... decimal was the tool of the past..

 

can you try your tool using a hex base?

 

16 digits before repeating?

 

throughly confused by the way.. my math skills don't go beyong typing in 111 then plus then 999 and jamming the repeater till i saw E.. but i never have, always blamed it on adult ADD

[alxian]

ok

 

you'd have to share this program though.. i've seen number paterns also (111+999) but the way you talk about the bases not having anything to do with the factors of an integer, i'd have to see it to understand it.