Special numbers
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Special Numbers. Harmonic Numbers. Perfect Numbers. PHI. Phi – The Phinest number around. This is the “Golden Ratio”. It can be derived from: Since n 2 -n 1 -n 0 =0 n 2 -n-1=0 n 2 =n+1 The root of which is 0.5(5 1/2 +1) Which can be approximated to:.

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Special Numbers

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Special numbers

Special Numbers

Harmonic Numbers

Perfect Numbers

PHI


Phi the phinest number around

Phi – The Phinest number around

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:


Phi to the first 1000 decimal places

Phi to the first 1000 decimal places

  • 1.618033988749894848204586834365638117720309179805762862135448622705260462818902449707207204189391137484754088075386891752126633862223536931793180060766726544333890865959395829056383226613199282902678806752087668925017116962070322210432162695486262963136144381497587012203408058879544547492461856953648644492410443207713449470495658467885098743394422125448770664780915884607499887124007652170575179788341662562494075890697040002812104276217711177780531531714101170466659914669798731761356006708748071013179523689427521948435305678300228785699782977834784587822891109762500302696156170025046433824377648610283831268330372429267526311653392473167111211588186385133162038400522216579128667529465490681131715993432359734949850904094762132229810172610705961164562990981629055520852479035240602017279974717534277759277862561943208275051312181562855122248093947123414517022373580577278616008688382952304592647878017889921990270776903895321968198615143780314997411069260886742962267575605231727775203536

Phee Phi Pho Phum

I smell the blood of a Mathematician


Special numbers

BUT WHY DOES THAT MATTER?!?!

Well…


Check this out

Check this out!

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

And you can see that this ratio appears everywhere!


In your face

IN YOUR FACE!!!


In nature

In nature


So what does it all mean

So what does it all mean?

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…

?


Perfect numbers

Perfect Numbers

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)


This starts going up very quickly

This starts going up very quickly

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


How to find a perfect number

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).

How to find a perfect number:


Nicomachus c 60 c 120

Nicomachus (c. 60 –c. 120)

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


The 4 th rule

The 4th rule

Take the example of when k=11:

210(211-1)=1024x2047=2096128

Therefore the 4th rule is also incorrect

Check again


5 th rule

5th rule

Can’t dispute it.

To date there are 39 known perfect numbers

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


Perfect harmony

Perfect Harmony

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


This sequence goes a little bit like this

This sequence goes a little bit like this:

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!


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