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Mathematics Dictionary
Dr. K. G. Shih
Senior High Mathematics
Z004 : Find 3rd perfect number



Questions


Q01. Find 2nd perfect number

Properties of perfect number.
  • 1. Perfect number = 1^3 + 3^3 + 5^3 + .... + (2*m -1)^3
  • 2. Perfect number = (2^(n-1))*(2^n-1).
  • 3. 2^n - 1 is a prime number for a perfect number
First perfect number
  • 1, 2, 3 are factors of 6
  • Sum factors 1 + 1 + 3 = 6
  • Hence 6 is the first perfect number
Find 2nd perfect number by rules
  • Rule 1 : P1 = 1 + 3^3 = 1 + 27 = 28
  • Rule 2 : P2 = (2^(3 - 1))*(2^3 - 1) = (2^2)*(8 - 1) = 28
  • Since P1 = P2
  • Hence 28 is the 2nd perfect number
Find 2nd perfect numbers by factors
  • Factors of 28 : 1, 2, 4, 7, 14
  • Sum of factors : P = 1 + 2 + 4 + 7 + 14 = 28

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Q02. Find 3rd perfect number

Properties of perfect number.
  • 1. Perfect number = 1^3 + 3^3 + 5^3 + .... + (2*m - 1)^3
  • 2. Perfect number = (2^(n-1))*(2^n-1).
  • 3. 2^n - 1 is a prime number for a perfect number
Find third perfect number using rules
  • Rule 1
    • m = 3 : P1 = 1 + 3^3 + 5^3 = 1 + 27 + 125 = 153
    • m = 4 : P1 = 1 + 3^3 + 5^3 + 7^3 = 153 + 343 = 496
  • Rule 2
    • n = 5 : P2 = (2^(5 - 1))*(2^5 - 1) = 16*31 = 496
  • Hence 496 is the 3rd perfect number
Find 3rd perfect number by factors
  • Factros of 496 : 1, 2, 4, 8, 16, 31, 62, 124, 248
  • Sum of Factros : 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248
  • Hence 496 is the 3rd perfect number

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Q03. Reference
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