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Special Numbers

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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  1. Special Numbers Harmonic Numbers Perfect Numbers PHI

  2. 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:

  3. Phi to the first 1000 decimal places • 1.618033988749894848204586834365638117720309179805762862135448622705260462818902449707207204189391137484754088075386891752126633862223536931793180060766726544333890865959395829056383226613199282902678806752087668925017116962070322210432162695486262963136144381497587012203408058879544547492461856953648644492410443207713449470495658467885098743394422125448770664780915884607499887124007652170575179788341662562494075890697040002812104276217711177780531531714101170466659914669798731761356006708748071013179523689427521948435305678300228785699782977834784587822891109762500302696156170025046433824377648610283831268330372429267526311653392473167111211588186385133162038400522216579128667529465490681131715993432359734949850904094762132229810172610705961164562990981629055520852479035240602017279974717534277759277862561943208275051312181562855122248093947123414517022373580577278616008688382952304592647878017889921990270776903895321968198615143780314997411069260886742962267575605231727775203536 Phee Phi Pho Phum I smell the blood of a Mathematician

  4. BUT WHY DOES THAT MATTER?!?! Well…

  5. Check this out! You can make a ruler based on this ratio looking like this: And you can see that this ratio appears everywhere!

  6. IN YOUR FACE!!!

  7. In nature

  8. 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… ?

  9. 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)

  10. 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

  11. 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:

  12. 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

  13. The 4th rule Take the example of when k=11: 210(211-1)=1024x2047=2096128 Therefore the 4th rule is also incorrect Check again

  14. 5th rule Can’t dispute it. To date there are 39 known perfect numbers The last of which is: 213466916(213466917 - 1).

  15. 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

  16. 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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