Strange Numbers

[alexander]

and i can go higher then 2145615872, but oh man does it take some time... eeh, no worries, i can factor quite arbitrarily large number in this manner...

 

if anyone knows of better factoring algorithm, (and i posted some before) please let me know!

[alexander]

Number to factor: 2145615872

Factors:

1

2

4

8

16

32

64

128

256

512

1024

2048

4096

8192

16384

32768

65479

130958

261916

523832

1047664

2095328

4190656

8381312

16762624

33525248

67050496

134100992

268201984

536403968

1072807936

 

Number to factor: 4860050

Factors:

1

2

5

10

13

25

26

50

65

130

325

650

7477

14954

37385

74770

97201

186925

194402

373850

486005

972010

2430025

 

Number to factor: 105664

Factors:

1

2

4

8

13

16

26

32

52

64

104

127

208

254

416

508

832

1016

1651

2032

3302

4064

6604

8128

13208

26416

52832

 

Number to factor: 170612

Factors:

1

2

4

13

17

26

34

52

68

193

221

386

442

772

884

2509

3281

5018

6562

10036

13124

42653

85306

 

Number to factor: 35019968

Factors:

1

2

4

8

16

32

64

131

262

524

1048

2096

4177

4192

8354

8384

16708

33416

66832

133664

267328

547187

1094374

2188748

4377496

8754992

17509984

 

Number to factor: 53032832

Factors:

1

2

4

8

16

32

64

128

317

634

1268

1307

2536

2614

5072

5228

10144

10456

20288

20912

40576

41824

83648

167296

414319

828638

1657276

3314552

6629104

13258208

26516416

[alexander]

What's more, the factor 4096 is 10 times the Perfect 496

did i miss something? 4096/496 = 8.2580645161

[alexander]

aah, oops, sorry, i follow that now :phones:

[CraigD]

[new]Conjecture: All odd powers of 2 are either a Cube of an odd power of 2, or the sum of a Perfect-Square-multiple of a Perfect number and a Perfect Square. [(Square*Perfect)+Square]

...

I have reached the limit of my hand calculator, and I'm ready for comment & criticism. :hihi: :cup: :cheer:

I’ll mull it over. I’ll try a dumb arbitrary precision calculator disproof until something smart comes of the mulling – we may get lucky, though a really dumb approach looks to be pretty computationally intense.

[Qfwfq]

Certainly, [imath]2^{2n}=(2^n)^2[/imath] and, also, the powers of two which are cubes are all and only those where the exponent is a multiple of 3:

 

[math]2^{3n}=(2^n)^3[/math]

[CraigD]

[new]Conjecture: All odd powers of 2 are either a Cube of an odd power of 2, or the sum of a Perfect-Square-multiple of a Perfect number and a Perfect Square. [(Square*Perfect)+Square] …

I’ll mull it over. I’ll try a dumb arbitrary precision calculator disproof until something smart comes of the mulling – we may get lucky, though a really dumb approach looks to be pretty computationally intense.

It took me more “smarts” to get “computationally intense” down to something manageable enough to produce even a small test. I need to revise my approach, as I’ve hit a computational bottleneck with my counting approach that’s preventing it from getting very high in a reasonable amount of time.

 

I checked the all [math]2^{2n+1}[/math] where [math]2n+1 \not= 3m[/math] (the first “or” condition of the conjecture) up to [math]2^{85}[/math], and found no counterexamples. Here are those odd powers of two in terms of

2^5=28*1^2+2^2
2^7=28*1^2+10^2
2^11=496*2^2+8^2
2^13=8128*1^2+8^2
2^17=8128*4^2+32^2
2^19=8128*8^2+64^2
2^23=8128*31^2+760^2
2^25=33550336*1^2+64^2
2^29=33550336*4^2+256^2
2^31=33550336*8^2+512^2
2^35=8589869056*2^2+512^2
2^37=137438691328*1^2+512^2
2^41=137438691328*4^2+2048^2
2^43=137438691328*8^2+4096^2
2^47=137438691328*32^2+16384^2
2^49=137438691328*64^2+32768^2
2^53=137438691328*256^2+131072^2
2^55=137438691328*512^2+262144^2
2^59=137438691328*2048^2+1048576^2
2^61=2305843008139952128*1^2+32768^2
2^65=2305843008139952128*4^2+131072^2
2^67=2305843008139952128*8^2+262144^2
2^71=2305843008139952128*32^2+1048576^2
2^73=2305843008139952128*64^2+2097152^2
2^77=2305843008139952128*256^2+8388608^2
2^79=2305843008139952128*512^2+16777216^2
2^83=2305843008139952128*2048^2+67108864^2
2^85=2305843008139952128*4096^2+134217728^2

There are lots of coincidences here – such as, with an exception at [math]2^{23}[/math], all the perfect squares being of powers of two – which I super-strongly suspect is due to the known perfect numbers having the form [math]2^{n-1}(2^n-1)[/math]. For example, consider:

[math]2^{85}=2305843008139952128 \cdot 4096^2 +134217728^2[/math]

[math]2^{85}=2305843008139952128 \cdot (2^{12})^2 +(2^{27})^2[/math]

[math]2^{85}=(2^{12})^2 \cdot ( 2305843008139952128 +(2^{15})^2)[/math]

[math]2^{85}=(2^{27})^2 \cdot (2147483647 +1)[/math]

[math]2^{85}=2^{54} \cdot ((2^{31}-1) +1)[/math]

[math]2^{85}=2^{54} \cdot 2^{31}[/math]

 

A proof seems not too far off :)

[CraigD]

[old]Conjecture: All odd powers of 2 are either a Cube of an odd power of 2, or the sum of a Perfect-Square-multiple of a Perfect number and a Perfect Square. [(Square*Perfect)+Square]

The conjecture can be simplified to:

All odd powers of 2 greater than 32 are at least one Perfect-Square-multiple of a Perfect number plus a Perfect Square.

 

We can prove this as follows:

[math]2^{2a+1} = 2^{n-1}(2^n-1)2^{2j} +2^{2k}[/math]

We are allowed to select [math]j[/math] and [math]k[/math] for a given [math]a[/math] and [math]n[/math], so may define them as follows:

[math]k = \frac{n-1 +2j}{2}[/math]

then

[math]2^{2a+1} = 2^{n-1}(2^n-1)2^{2j} +2^{n-1 +2j}[/math]

[math]2^{2a+1} = 2^{n-1 +2j}(2^n-1+1)[/math]

[math]2^{2a+1} = 2^{2n-1 +2j}[/math]

giving

[math]2a +1 = 2n -1 +2j[/math]

so

[math]j = a -n +1[/math]

[math]k = \frac{2a +1 –n}2[/math]

Have positive integer values for any odd value of [math]n[/math]

 

According to a proof by Euler, [math]2^{n-1}(2^n-1)[/math] is a perfect number if [math]2^n-1[/math] is prime (a Mersenne prime). So the conjecture holds for any of the known perfect numbers greater than [math]6[/math] ([math]2^{2-1}(2^2-1)[/math]).

 

We can use this to quickly generate lists like this

2^7=(2^2)(2^3-1)((2^1)^2)+(2^2)^2
2^9=(2^2)(2^3-1)((2^2)^2)+(2^3)^2
2^11=(2^4)(2^5-1)((2^1)^2)+(2^3)^2
2^13=(2^4)(2^5-1)((2^2)^2)+(2^4)^2
2^15=(2^6)(2^7-1)((2^1)^2)+(2^4)^2
2^17=(2^6)(2^7-1)((2^2)^2)+(2^5)^2
2^19=(2^6)(2^7-1)((2^3)^2)+(2^6)^2
2^21=(2^6)(2^7-1)((2^4)^2)+(2^7)^2
2^23=(2^6)(2^7-1)((2^5)^2)+(2^8)^2
2^25=(2^6)(2^7-1)((2^6)^2)+(2^9)^2
2^27=(2^12)(2^13-1)((2^1)^2)+(2^7)^2
2^29=(2^12)(2^13-1)((2^2)^2)+(2^8)^2
2^31=(2^12)(2^13-1)((2^3)^2)+(2^9)^2
2^33=(2^12)(2^13-1)((2^4)^2)+(2^10)^2
2^35=(2^16)(2^17-1)((2^1)^2)+(2^9)^2
2^37=(2^16)(2^17-1)((2^2)^2)+(2^10)^2
2^39=(2^18)(2^19-1)((2^1)^2)+(2^10)^2
2^41=(2^18)(2^19-1)((2^2)^2)+(2^11)^2
2^43=(2^18)(2^19-1)((2^3)^2)+(2^12)^2
2^45=(2^18)(2^19-1)((2^4)^2)+(2^13)^2
2^47=(2^18)(2^19-1)((2^5)^2)+(2^14)^2
2^49=(2^18)(2^19-1)((2^6)^2)+(2^15)^2
2^51=(2^18)(2^19-1)((2^7)^2)+(2^16)^2
2^53=(2^18)(2^19-1)((2^8)^2)+(2^17)^2
2^55=(2^18)(2^19-1)((2^9)^2)+(2^18)^2
2^57=(2^18)(2^19-1)((2^10)^2)+(2^19)^2
2^59=(2^18)(2^19-1)((2^11)^2)+(2^20)^2
2^61=(2^18)(2^19-1)((2^12)^2)+(2^21)^2
2^63=(2^30)(2^31-1)((2^1)^2)+(2^16)^2
2^65=(2^30)(2^31-1)((2^2)^2)+(2^17)^2
2^67=(2^30)(2^31-1)((2^3)^2)+(2^18)^2
2^69=(2^30)(2^31-1)((2^4)^2)+(2^19)^2
2^71=(2^30)(2^31-1)((2^5)^2)+(2^20)^2
2^73=(2^30)(2^31-1)((2^6)^2)+(2^21)^2
2^75=(2^30)(2^31-1)((2^7)^2)+(2^22)^2
2^77=(2^30)(2^31-1)((2^8)^2)+(2^23)^2
2^79=(2^30)(2^31-1)((2^9)^2)+(2^24)^2
2^81=(2^30)(2^31-1)((2^10)^2)+(2^25)^2
2^83=(2^30)(2^31-1)((2^11)^2)+(2^26)^2
2^85=(2^30)(2^31-1)((2^12)^2)+(2^27)^2
2^87=(2^30)(2^31-1)((2^13)^2)+(2^28)^2
2^89=(2^30)(2^31-1)((2^14)^2)+(2^29)^2
2^91=(2^30)(2^31-1)((2^15)^2)+(2^30)^2
2^93=(2^30)(2^31-1)((2^16)^2)+(2^31)^2
2^95=(2^30)(2^31-1)((2^17)^2)+(2^32)^2
2^97=(2^30)(2^31-1)((2^18)^2)+(2^33)^2
2^99=(2^30)(2^31-1)((2^19)^2)+(2^34)^2

, or construct some spectacularly large example, such as

[math]2^{65165315}= 2^{32582657}(2^{32582656}-1) \cdot 2^2 + (2^{8145665})^2[/math], a 9808358 digit decimal numeral.

 

Note that, for values larger than [math]2^9[/math], solutions in this form aren’t unique. For example:

2^7=(2^2)(2^3-1)((2^1)^2)+(2^2)^2
2^9=(2^2)(2^3-1)((2^2)^2)+(2^3)^2
2^11=(2^4)(2^5-1)((2^1)^2)+(2^3)^2
2^11=(2^2)(2^3-1)((2^3)^2)+(2^4)^2
2^13=(2^4)(2^5-1)((2^2)^2)+(2^4)^2
2^13=(2^2)(2^3-1)((2^4)^2)+(2^5)^2
2^15=(2^6)(2^7-1)((2^1)^2)+(2^4)^2
2^15=(2^4)(2^5-1)((2^3)^2)+(2^5)^2
2^15=(2^2)(2^3-1)((2^5)^2)+(2^6)^2
2^17=(2^6)(2^7-1)((2^2)^2)+(2^5)^2
2^17=(2^4)(2^5-1)((2^4)^2)+(2^6)^2
2^17=(2^2)(2^3-1)((2^6)^2)+(2^7)^2
2^19=(2^6)(2^7-1)((2^3)^2)+(2^6)^2
2^19=(2^4)(2^5-1)((2^5)^2)+(2^7)^2
2^19=(2^2)(2^3-1)((2^7)^2)+(2^8)^2
2^21=(2^6)(2^7-1)((2^4)^2)+(2^7)^2
2^21=(2^4)(2^5-1)((2^6)^2)+(2^8)^2
2^21=(2^2)(2^3-1)((2^8)^2)+(2^9)^2
2^23=(2^6)(2^7-1)((2^5)^2)+(2^8)^2
2^23=(2^4)(2^5-1)((2^7)^2)+(2^9)^2
2^23=(2^2)(2^3-1)((2^9)^2)+(2^10)^2
2^25=(2^6)(2^7-1)((2^6)^2)+(2^9)^2
2^25=(2^4)(2^5-1)((2^8)^2)+(2^10)^2
2^25=(2^2)(2^3-1)((2^10)^2)+(2^11)^2
2^27=(2^12)(2^13-1)((2^1)^2)+(2^7)^2
2^27=(2^6)(2^7-1)((2^7)^2)+(2^10)^2
2^27=(2^4)(2^5-1)((2^9)^2)+(2^11)^2
2^27=(2^2)(2^3-1)((2^11)^2)+(2^12)^2
2^29=(2^12)(2^13-1)((2^2)^2)+(2^8)^2
2^29=(2^6)(2^7-1)((2^8)^2)+(2^11)^2
2^29=(2^4)(2^5-1)((2^10)^2)+(2^12)^2
2^29=(2^2)(2^3-1)((2^12)^2)+(2^13)^2
2^31=(2^12)(2^13-1)((2^3)^2)+(2^9)^2
2^31=(2^6)(2^7-1)((2^9)^2)+(2^12)^2
2^31=(2^4)(2^5-1)((2^11)^2)+(2^13)^2
2^31=(2^2)(2^3-1)((2^13)^2)+(2^14)^2
2^33=(2^12)(2^13-1)((2^4)^2)+(2^10)^2
2^33=(2^6)(2^7-1)((2^10)^2)+(2^13)^2
2^33=(2^4)(2^5-1)((2^12)^2)+(2^14)^2
2^33=(2^2)(2^3-1)((2^14)^2)+(2^15)^2

[CraigD]

Given your proof, this is a full-fledged theorem now. Do you think it is likely already known? If not, do I get to name it?

Why not? Of course, unless it’s published in a good journal of two, and discussed a lot, hypography readers may be the only people who ever know of it, but I’d say we’ve gone thought the right motions to claim if for our own

 

Proceeding as it does from Euler’s proof of the mapping of perfect numbers to Mersenne primes, I’d rank it more as a sort of corollary to either his or Mersenne’s 18th century work. As I can’t right off think of any major direction suggested by it, I’m afraid I’d rank it a very minor corollary. Given how much Euler wrote, I wouldn’t be surprised if he jotted down something much like it. I doubt anyone but a very knowledgeable math history scholar specializing in Euler would know of such a thing.

 

A really deep insight into the relationship between prime factorizations – of which the powers of two are a special case of a number with a single factor, 2 – and addition would be very major. It could potentially overturn the bulk of modern cryptography, compromising the strength of algorithms like RSA. It could also open deep, deep paths into questions relating to the fundamental nature of everything! (insert suitable dramatic narrative sound effect)

 

This little theorem, however, is not such an insight.

[freeztar]

I'd definitely reference this thread. Also, you can protest the deletion with some good reasoning. :)

[Essay]

Having read the first page of this thread last weekend, I now recall thinking that you should submit an article to some math journal.

Now more than ever.... Do you live near a university? A librarian could probably help you with getting the guidelines for submission.

 

Some suggestions of journals that might be interested.

 

American Journal of Mathematics 0002-9327

American Mathematical Monthly, The 0002-9890

Annals of Operations Research 0254-5330

Canadian journal of mathematics 0008-414X

Combinatorica 0209-9683

Communications in algebra 0092-7872

Compositio mathematica 0010-437X

Computational complexity 1016-3328

Fundamenta mathematicae 0016-2736

Designs, codes, and cryptography 0925-1022

Integers 1553-1732

Izvestiya. Mathematics 1064-5632

Journal of integer sequences 1530-7638

Journal of number theory 0022-314X

Journal of Numerical Mathematics 1570-2820

Lithuanian mathematical journal 0363-1672

Mathematics Magazine 0025-570X

Mathematics news letter 1539-557X

Michigan mathematical journal 0026-2285

Missouri journal of mathematical sciences 1085-2581

Numerical algorithms 1017-1398

Numerische Mathematik 0029-599X

Pattern recognition letters 0167-8655

SIAM journal on numerical analysis 0036-1429

 

I think what you've got is neat, but I have to agree with wiki that there needs to be some independant verification.

At the least, maybe some other forums or math blogs.

Is there a math page (oddities or patterns) on wiki that you could piggyback onto? Including a discussion (behind the article) can be helpful to getting an edit or addition accepted.

:)

[CraigD]

I think we’d best support the deletion of the wikipedia article on strange numbers.

 

Wikipedia, or any encyclopedia, is intended to provide a reference to terms that a person may encounter when reading material that doesn’t explain them fully. Strange and bizarre numbers exist, AFAIK, only on hypography, and are well explained there. The terms are really, IMHO, “working terms” invented to permit the concepts to be discussed at hypography.

 

If the ideas began being discussed more widely, at other internet forums or in traditional academic settings, they’re be some value in a wikipedia page, providing a history pointing back to hypography, Turtle, and the rest of us who’ve contributed to the thread over the years.

 

Here’s my acid test: if you google “strange numbers”, the first relevant hit (the use of the term to describe a collection of numbers) is this hypography thread. So, if someone wanted to know if the term “strange number” had ever been used to describe a collection of numbers, they’d be directed straight to the authoritative source.

 

Once a search begins showing lots of papers and other forums referring to “strange numbers”, there’s value to be had in an encyclopedia entry. ‘Til that happens, a wikipedia entry would only get in the way of the hypography thread.

[freeztar]

The first sentence threw me for a loop, but the reasoning following was reasonable. :)

 

I agree with what you've stated, Craig, and you make a good case against the "strange numbers" wiki page. Perhaps it does not need its own wiki page, but could it find a home within an existing wiki page? :shrug:

[CraigD]

TPerhaps it does not need its own wiki page, but could it find a home within an existing wiki page? :shrug:

A link from the wikipedia article for “abundant number”, along with some well-written text on the idea of subsets of the abundant numbers, would be OK, I think.

 

Another possibility is the OEIS (Sloan’s), which has an online submission form.

 

A journal article would still be good. Unfortunately, I've not written one in over 25 years, so am not up to speed with the process.

[Qfwfq]

A few bits of advice:

• I would say "defined by" rather than "discovered by" R. T. G.
  
• Mathematics isn't physics: Strange numbers don't need independent verification. It's a matter of whether the construct is free of logical errors and inconsistencies.
  
• Wiki isn't a place to write about one's own work (perhaps including a group of people), I'd say their concern is whether the terminology is generally used by mathematicians, or at least by enough to write along the lines of: "Some people define Strange Numbers as being..."
  
• The important thing to make it an intreresting mathematical construct is to lay the groundwork of what properties these numbers have, anything that can be proven consequently to the basic definitions etc.
  

 

Toward the last item in this list, perhaps all relevant things around here should be condensed in an orderly manner and then worked on. It would be a good idea for one fine volunteer :D to do this, sparing the necessity of others sifting through the great volume and summarizing.

:)

[Qfwfq]

how do I add another subscript to index individual elements in a set? [tag]S_n_m[/tag] gives a syntax error. :eek:

 

[math]S_n,m[/math] :doh:

S_{nm\rm\;and\;whatever\;the\;heck\;ya\;want!} for:

 

[math]S_{nm\rm\;and\;whatever\;the\;heck\;ya\;want!}[/math]

[CraigD]

now how do I add another subscript to index individual elements in a set? [tag]S_n_m[/tag] gives a syntax error. :eek:

{} are pretty key to LaTeX. Basically, you need to put {} around anything longer than 1 character that gets rendered by an operator or function (eg: _).

 

Try something like [math]A_{B_{n+1} , C}[/math].