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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This is the “Golden Ratio”.
It can be derived from:
The root of which is 0.5(51/2+1)
Which can be approximated to:
Phee Phi Pho Phum
I smell the blood of a Mathematician
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
(The first records of these came from Euclid around 300BC)
As you can see:
6, 28, 496, 8128,33550336, 8589869056,
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
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 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?
Take the example of when k=11:
Therefore the 4th rule is also incorrect
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
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!