Strange Numbers

[Tim_Lou]

these are indeed very interesting stuff, i can see how these strange (and all the others) are in the froms of p*q

 

but the anomalies really got me thinking... actually, i've been trying to figure out the pattern these couple days. i'll let you know if i find anything interesting. I'm no expert in this area, so don't expect too much....

[TheBigDog]

Tim has me thinking (:eek:) about the Mersenne Prime business & it jogged my memory on a discussion with BigDog. That is we talked about adding a feature to the Numberator that displays each element's Prime Factorization. Seeing that information allows us to quickly look for 2^k -1 where k is prime & a whole host of other patterns we have yet to see.

 

Bring On The Synergy!

/forums/images/smilies/banana_sign.gif

I put the order into the kitchen. It will be out shortly. Can I get you a beverage while you are waiting?

 

Bill

[TheBigDog]

these are indeed very interesting stuff, i can see how these strange (and all the others) are in the froms of p*q

 

but the anomalies really got me thinking... actually, i've been trying to figure out the pattern these couple days. i'll let you know if i find anything interesting. I'm no expert in this area, so don't expect too much....

I welcome your input Tim_Lou. I am no expert either. I am compelled by the puzzle of it. The rest comes naturally. :eek:

 

Bill

[TheBigDog]

Here you are, Mr Turtle.

 

Bill

[TheBigDog]

If anyone else is interested in running the Numberator please shoot me an e-mail. I will send you instructions for making it go.

 

Bill

[Dark Mind]

And Turtle's deleting posts again to keep his palindrome :shrug:...

 

:)

[TheBigDog]

And Turtle's deleting posts again to keep his palindrome :)...

 

:cup:

Of course he is! For these reasons...

 

k(4114)base 14 = 6 = first perfect number.

k(114)base 14 = A or 10

k(114)base 10 = 6 = first pefect number.

 

And the posts have the same digits as the threads, plus 4000.

 

Plus the palandrome. This is a significant combination of numbers and cannot be spent for the next without taking some time to take notice.

 

And Turtle, back to our conversation here...

 

I was looking through the whole set for numbers with 13 as a factor. There is only one in the shole set of nearly 7.4 million strange and naomalous numbers. It has me thinking that there three groups of prime numbers in these sets. "Root primes" define a strange set. "Shared primes" are foiund in various sets, but are rare. And "Unique primes" are found only one time in the whole set. By finnding a pattern to these we may unlock the math behind some of this.

 

Bill

[TheBigDog]

Yes, I had done a poor search. I searched only one set, not all of them. And that is why I found only one results. Upon further reflection this is what I have come up with.

 

Strange Root Primes = (2^1) (3^1) (all primes^1) The exception in the strange set is 24 which is (2^3) (3^1) The anomalies MAY be in a pattern, but it still eludes me with only 3 samples. They are:

 

304 = (2^4) (19^1)

127744 = (2^8) (499^1)

33501184 = (2^12) (8179^1)

 

Hpothetically the next one might be (2^16) (x^1) but this is really just a guess.

 

None of the anaomalies have more than four prime factors. In fact the whole set has 2, 3 or 4 prime factors.

 

There anomalies each share a prime factor with one of the numbers in the regular same set. Example: 114 Strange and 304 Strange anomaly both have 19 as a factor. Those are the only two places in the set where you find 19 as a factor. 2994 Strange and 127744 Strange Anomaly both have 499 as a factor. It is the only place where 499 is found in the strange set. 1988 Bizarrr and 4544 Bizarre anomaly both have 71 as a factor, and it is not found anyplace else in the set.

 

Using the Root Primes I successfully found numbers that are beyonnd our set. Although not any more anomalies. They remain puzzling.

 

Bill

[TheBigDog]

UPDATE: The PC dedicated to running the numberator rebooted on 5/31 when active directory pushed out MS updates over the network. I restarted it this morning. It is almost to 815,000,000. No new anomalies since the last snapshot at 615,000,000. Could they end!?!? :D :lol: :hihi:

 

Bill

[Tim_Lou]

nevermind the nevermind,

yeah! the theorem is done! check my next post!!!!!

[TheBigDog]

Two more Anomalies! 850,310,432 and 855,935,072! It skipped over 325,000,000 befre it came to these two numbers.

 

Bill

[Tim_Lou]

:evil: :evil: NOW, BE READY, AND EMBRACE, THE FIRST THEOREM ORIGINATED FROM http://www.hypography.com

 

This is the theorem explaining the anomalies in the form of [math]2^{k-1}a[/math] where a is a prime.

 

with the help of many others, we made it!!!!!!!

i have yet to proofread it, there might be some errors... hopfully the logic is correct!

 

now there it is!!!!

I am so excited!!!!!!!!!!!!

tell me what you think about it.

 

i think that in the proving process, the converse of the theorem is also proved. I'm not sure though.

 

if it is correct, i hope you guys can come up with a name for that theorem. (make it cool)

 

hmm... hopfully, as time goes on, i will be able to come up with an explaination of all the anomalies!!! :eek: :hihi: :hihi:

 

edit: fixed some minor errors and typos in the document

 

june 06 11:40 pm

edit: fixed another MAJOR typo in the statement of the theorem. Gosh, how could i have overseen that TYPO!!!! the theorem said 2^(k-1)a is abundant...bla bla bla but it never specifies what a is!!!! dOh!!!

Anomalies in the form of 2^ka fixed.pdf

[Tim_Lou]

predictions of the theorem:

anomalies:

[math]2^{12}*(2^2-2^4+2^{13}-1)[/math]

[math]2^{16}*(2^2-2^4+2^{17}-1)[/math]

[math]2^{56}*(2^2-2^4+2^{57}-1)[/math]

[math]2^{104}*(2^2-2^4+2^{105}-1)[/math]

[math]2^{136}*(2^2-2^4+2^{137}-1)[/math]

 

they are all strange numbers anomalies, abundant by 12

[Tim_Lou]

jesus, i cannot tell you how excited i AM!!!!!

this is just like the time when i first solved the 2-body problem!!!!!!!

I'm just!!!!

Oh, man i'm dancing and yelling all around my room... we still got plenty of work to do though!!!!!!! hopfully, theBigDog will come online soon and we will have a discussion about the theorem!

[TheBigDog]

He head spins! It sure shootin finds the strange anomalies! At least as far as I can calculate. I need to make my massive number calculator addition to the Numberator. That will allow us to go unlimited in out quest! The order is in the kitchen. The cook says he needs to shop for ingredients. Sent out some more breadsticks in the mean time. :)

 

Bill

[TheBigDog]

Acknowledged #1; a perfect reply if I've ever seen one. Now as you all may know from friend Aesop, I have some experience in contesting with hares. Just so I need to help friend Tim come up to speed with some other theoretical work accomplished here at Hypography over the past year and a half on this very topic. Threads we left:

 

http://hypography.com/forums/physics-mathematics/1410-statisitical-view-integers-grouped-number-divisors.html?highlight=Phat+number

 

http://hypography.com/forums/physics-mathematics/1473-big-r-hunt-phat-numbers.html?highlight=Phat+number

 

So thens Mr. Bill; no exceptions to our list of anomalies & Mr. Tim's expression? Fascinating! :)

Strange anomalies only. Bizarre, curious, quirky, unusual, etc remain at large.

 

Plunk 8589082624 into the Numberator and see...

 

Bill

[Tim_Lou]

the theorem is for all anomalies in the form of 2^k*a

some more examples are:

abundant by 56:

[math]2^{5}*(2^3-2^6+2^{6}-1)[/math]

[math]2^{6}*(2^3-2^6+2^{7}-1)[/math]

[math]2^{7}*(2^3-2^6+2^{8}-1)[/math]

[math]2^{9}*(2^3-2^6+2^{10}-1)[/math]

[math]2^{15}*(2^3-2^6+2^{16}-1)[/math]

[math]2^{18}*(2^3-2^6+2^{19}-1)[/math]

[math]2^{21}*(2^3-2^6+2^{22}-1)[/math]

[math]2^{27}*(2^3-2^6+2^{28}-1)[/math]

[math]2^{42}*(2^3-2^6+2^{43}-1)[/math]

 

abundant by 992

[math]2^{9}*(2^5-2^{10}+2^{10}-1)[/math]

[math]2^{13}*(2^5-2^{10}+2^{14}-1)[/math]

[math]2^{16}*(2^5-2^{10}+2^{17}-1)[/math]

[math]2^{25}*(2^5-2^{10}+2^{26}-1)[/math]

[math]2^{28}*(2^5-2^{10}+2^{29}-1)[/math]

[math]2^{53}*(2^5-2^{10}+2^{54}-1)[/math]

 

well, my calculator is too slow for the bigger ones... but you get the idea...

 

actually, perhaps BigDog can make a program to generate these numbers!

firstly, find a list of mersenne prime [math]2^n-1[/math]

for each n, iterate the integer k through 1 to some bound and see if

[math]2^2-2^{2n}+2^k-1[/math] is prime. if it is, then

[math]2^{k-1}*(2^2-2^{2n}+2^k-1)[/math] is an anomaly.

as for which anomaly, simply calculate [math]2^p*(2^p-1)[/math]

 

by the way, do you think the theorem has ever been discovered before? if not, let us get a cool name for it!!!! let's name the theorem!

 

now, my next project will be the anomalies in the form of 2^k*a*b.... well good luck to me. (I am still scratching my head as to how to approach this problem)

 

edit: added some more the the bizarre numbers.

 

again, if you want to examine the proof and the theorem, check out page 18, post #177