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Special Numbers

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Harmonic Numbers

Perfect Numbers

PHI

This is the “Golden Ratio”.

It can be derived from:

Since n2-n1-n0=0

n2-n-1=0

n2=n+1

The root of which is 0.5(51/2+1)

Which can be approximated to:

- 1.618033988749894848204586834365638117720309179805762862135448622705260462818902449707207204189391137484754088075386891752126633862223536931793180060766726544333890865959395829056383226613199282902678806752087668925017116962070322210432162695486262963136144381497587012203408058879544547492461856953648644492410443207713449470495658467885098743394422125448770664780915884607499887124007652170575179788341662562494075890697040002812104276217711177780531531714101170466659914669798731761356006708748071013179523689427521948435305678300228785699782977834784587822891109762500302696156170025046433824377648610283831268330372429267526311653392473167111211588186385133162038400522216579128667529465490681131715993432359734949850904094762132229810172610705961164562990981629055520852479035240602017279974717534277759277862561943208275051312181562855122248093947123414517022373580577278616008688382952304592647878017889921990270776903895321968198615143780314997411069260886742962267575605231727775203536

Phee Phi Pho Phum

I smell the blood of a Mathematician

BUT WHY DOES THAT MATTER?!?!

Well…

You can make a ruler based on this ratio looking like this:

And you can see that this ratio appears everywhere!

Some people take this to be a proof that god exists as all of these things could not be based on this same ratio purely by chance. This suggests a creator or designer…

?

A perfect number is a positive integer which is the sum of all it’s positive divisors (e.g. 6 being the sum of 1, 2 and 3)

The first 4 perfect numbers are 6, 28, 496 and 8128

1+2+3=6

1+2+4+7+14=28

1+2+4+8+16+31+62+124+248=496

1+2+4+8+16+32+64+127+254+508+1016+2032+4064=8128

(The first records of these came from Euclid around 300BC)

As you can see:

6, 28, 496, 8128,33550336, 8589869056,

137438691328, 2305843008139952128,

2658455991569831744654692615953842176,

191561942608236107294793378084303638130997321548169216,

13164036458569648337239753460458722910223472318386943117783728128

You can take my word for it or if you want you can work them out. =P

According to Euclid, if you start with 1 and keep adding the double of the number preceding it until the sum is a prime number

e.g. 1+2+4=7

Then take the last number (4) and the sum (7) then you should get a perfect number 4x7=28

Also from 1+2+3+4…+2k-1=2k-1

We can rearrange to 2k-1(2k-1) should be a perfect number (so long as 2k-1 is prime).

Nicomachus added some extra rules for perfect numbers:

1.)The nth perfect number has n digits.2.) All perfect numbers are even.3.) All perfect numbers end in 6 and 8 alternately.

4.) Euclid's algorithm to generate perfect numbers will give all perfect numbers i.e. every perfect number is of the form 2k-1(2k - 1), for some k > 1, where 2k - 1 is prime.5.) There are an infinite amount of perfect numbers.

At this time however, only the first 4 perfect numbers had been found, do these rules apply to the rest of them?

Check

Take the example of when k=11:

210(211-1)=1024x2047=2096128

Therefore the 4th rule is also incorrect

Check again

Can’t dispute it.

To date there are 39 known perfect numbers

The last of which is: 213466916(213466917 - 1).

Perfect numbers are all thought to be Harmonic numbers integer whose divisors have a harmonic mean that is an integer.

e.g. 6 which has the divisors 1, 2, 3 and 6

And 140:

=5

1, 6, 28, 140, 270, 496, 672, 1638, 2970, 6200, 8128, 8190 …

Including the perfect numbers: 6, 28, 496, 8128

However: This could also be as wrong as Nicomachus so beware!