Perfect Numbers: An Aural Dissertation

[Turtle]

On to the next: a solution for n where 2n^2-n=8,589,869,056

 

:circle:

 

Found it! Found them I mean. By-the-by I am using trial & error and focussing on even powers of 2 for testing.

The solution for n where 2n^2-n=8,589,869,056 is 65,536 [2^16]

The solution for n where 2n^2-n=137,438,691,328 is 262,144 [2^18]

 

About to go Cheshire Cat,

Turtle

[Turtle]

Look as I may, I find no example of my formula for perfect numbers extant. Here's the standard song & dance:

 

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

 

All of the perfect numbers that have been found so far fit the formula

 

 

2^(n-1) * ( 2^n - 1 )

 

where "n" is one of a very short list of prime numbers that can be used to create "Mersenne" prime numbers. You will notice that the eight numbers I found match this formula:

2^1 * ( 2^2 - 1 ) = 2 * 3 = 6

2^2 * ( 2^3 - 1 ) = 4 * 7 = 28

2^4 * ( 2^5 - 1 ) = 16 * 31 = 496

2^6 * ( 2^7 - 1 ) = 64 * 127 = 8,128

2^12 * (2^13-1) = 4096 * 8191 = 33,550,336

2^16 * (2^17-1) = 65536 * 131071 = 8,589,869,056

2^18 * (2^19-1) = 262144 * 524287 = 137,438,691,328

2^30 * (2^31-1) = 1073741824*2147483647=2,305,843,008,139,952,128

 

The first 35 perfect numbers fit this same formula with "n" values of:

 

2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607,

1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213,

19937, 21701, 23209, 44497, 86243, 110503, 132049,

216091, 756839, 859433, 1257787, and 1398269.

 

 

What I found:

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=262,144 :N=33,550,336

n=65,536 :N= 8,589,869,056

n=262,144 :N=137,438,691,328

 

Inasmuch as 2n^2-n is the expression for 6-sided numbers in the class of Figurate numbers, I (again) make the conjecture that all Perfect Numbers are Hexagonal. Who ya gonna call? :shrug:

[Turtle]

Inasmuch as 2n^2-n is the expression for 6-sided numbers in the class of Figurate numbers, I (again) make the conjecture that all Perfect Numbers are Hexagonal. Who ya gonna call? :cup:

Right...so I called on Google with a search of "worlds foremost number theorist" & I came up with Robert P. Langlands School of Mathematics Institute for Advanced Study Princeton N.J.. I have sent Robert a communique del la electronique asking if my observation/discovery is indeed new & directing him to this thread at Hypogrpahy.

Kaffee bitte?:cup: :cup:

[TheBigDog]

Right...so I called on Google with a search of "worlds foremost number theorist" & I came up with Robert P. Langlands School of Mathematics Institute for Advanced Study Princeton N.J.. I have sent Robert a communique del la electronique asking if my observation/discovery is indeed new & directing him to this thread at Hypogrpahy.

Kaffee bitte?:cup: :cup:

The LargeNumberCalculator is nearly done fermenting. You will be able to test perfect numbers bigger than your imagination!

 

Bill

[Turtle]

Look as I may, I find no example of my formula for perfect numbers extant.

 

What I found:

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=262,144 :N=33,550,336

n=65,536 :N= 8,589,869,056

n=262,144 :N=137,438,691,328

 

Inasmuch as 2n^2-n is the expression for 6-sided numbers in the class of Figurate numbers, I (again) make the conjecture that all Perfect Numbers are Hexagonal. Who ya gonna call? :)

 

I put in a call to mathworld!

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

 

...In addition, all even perfect numbers are hexagonal numbers, so it follows that even perfect numbers are always the sum of consecutive positive integers starting at 1, ...

My question answered & a new tidbit ta boot. :ud:

[Turtle]

I put in a call to mathworld!

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

...In addition, all even perfect numbers are hexagonal numbers, so it follows that even perfect numbers are always the sum of consecutive positive integers starting at 1, ...

 

My question answered & a new tidbit ta boot. :hihi:

 

This requires further elucidation. I clipped that quote because the example used summation notation that did not transfer through cut & paste. As I don't know the LATEX stuff yet, I'll write this out longhand.

 

In addition, all even perfect numbers are hexagonal numbers, so it follows that even perfect numbers are always the sum of consecutive positive integers starting at 1, for example,

 

6 = sum n for n= 1 to 3

28 = sum n for n = 1 to 7

496 = sum n for n = 1 to 31

.

.

.

(Singh 1997), where 3, 7, 31, ... (Sloane's A000668) are simply the Mersenne primes.

 

I added the red for emphasis of the Mersenne primes, and the blue for empahsis of the careless kind of talk that IMHO doesn't belong. There is no such thing as "simply" Mersenne.:doh:

 

What I find of particular interest now is having a look at this relationship between Mersenne primes & hexagonal numbers. We can take this up in the Katabatak thread where I am investigating the set of hexagonal numbers & their meta-set the Figurate numbers.

Intricater & intricater.:eek2: :cup: