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Special Numbers PHI Perfect Numbers Harmonic Numbers

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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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Phi to the first 1000 decimal places 1.618033988749894848204586834365638117720309179805762862 13544862270526046281890244970720720418939113748475408807 53868917521266338622235369317931800607667265443338908659 59395829056383226613199282902678806752087668925017116962 07032221043216269548626296313614438149758701220340805887 95445474924618569536486444924104432077134494704956584678 85098743394422125448770664780915884607499887124007652170 57517978834166256249407589069704000281210427621771117778 05315317141011704666599146697987317613560067087480710131 79523689427521948435305678300228785699782977834784587822 89110976250030269615617002504643382437764861028383126833 03724292675263116533924731671112115881863851331620384005 22216579128667529465490681131715993432359734949850904094 76213222981017261070596116456299098162905552085247903524 06020172799747175342777592778625619432082750513121815628 55122248093947123414517022373580577278616008688382952304 59264787801788992199027077690389532196819861514378031499 7411069260886742962267575605231727775203536 1.618033988749894848204586834365638117720309179805762862 13544862270526046281890244970720720418939113748475408807 53868917521266338622235369317931800607667265443338908659 59395829056383226613199282902678806752087668925017116962 07032221043216269548626296313614438149758701220340805887 95445474924618569536486444924104432077134494704956584678 85098743394422125448770664780915884607499887124007652170 57517978834166256249407589069704000281210427621771117778 05315317141011704666599146697987317613560067087480710131 79523689427521948435305678300228785699782977834784587822 89110976250030269615617002504643382437764861028383126833 03724292675263116533924731671112115881863851331620384005 22216579128667529465490681131715993432359734949850904094 76213222981017261070596116456299098162905552085247903524 06020172799747175342777592778625619432082750513121815628 55122248093947123414517022373580577278616008688382952304 59264787801788992199027077690389532196819861514378031499 7411069260886742962267575605231727775203536 Phee Phi Pho Phum I smell the blood of a Mathematician

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BUT WHY DOES THAT MATTER?!?! Well…

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

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IN YOUR FACE!!!

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In nature

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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… ?

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Perfect Numbers A perfect number is a positive integer which is the sum of all its 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)

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This starts going up very quickly As you can see: 6, 28, 496, 8128,33550336, 8589869056, 137438691328, 2305843008139952128, 2658455991569831744654692615953842176, 19156194260823610729479337808430363813 0997321548169216, 13164036458569648337239753460458722910 223472318386943117783728128 You can take my word for it or if you want you can work them out. =P

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

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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 2 k-1 (2k - 1), for some k > 1, where 2 k - 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

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The 4 th rule Take the example of when k=11: 2 10 (2 11 -1)=1024x2047=2096128 Therefore the 4 th rule is also incorrect Check again

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5 th rule Cant dispute it. To date there are 39 known perfect numbers The last of which is: 2 13466916 (2 13466917 - 1).

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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

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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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Slide 5 - 1 Copyright © 2009 Pearson Education, Inc. Slide 5 - 1 Copyright © 2009 Pearson Education, Inc. Chapter 1 Number Theory and the Real Number System.

Slide 5 - 1 Copyright © 2009 Pearson Education, Inc. Slide 5 - 1 Copyright © 2009 Pearson Education, Inc. Chapter 1 Number Theory and the Real Number System.

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