Perfect number

What is a Perfect number?

• A perfect number is a positive integer that is equal to the sum of its proper divisors (factors) excluding itself is called a perfect number.
• The smallest perfect number is 6, which is the sum of 1, 2, and 3. Other perfect numbers are 28, 496, and 8128.

6 = 1 + 2 + 3,
28 = 1 + 2 + 4 + 7 + 14,
496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248
8128 = 1 + 2 + 4 + 8 + 16 + 32 + 64 + 127 + 254 + 508 + 1016 + 2032 + 4064

Euclid’s algorithm to generate perfect numbers

• The first recorded mathematical result about perfect numbers is known as Euclid‘s Elements written by Greek Mathematician Euclid; around 300BC
• Euclid’s algorithm to generate perfect numbers:

N= 2^p-1(2^p – 1)

Where : p is a prime number

Mersenne prime numbers

• Marin Mersenne was a French monk who is best known for his role as a clearing house for correspondence between eminent philosophers and scientists and for his work in number theory
  

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• A  Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form M_n = 2ⁿ – 1
  

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• The primes are 2, 3, 5, 7, 13, 17, 19, 31, … resulting Mersenne primes are 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, ……
  

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  • For p= 11 , 2¹¹ − 1 = 2047 = 23 × 89 therefore 2047 is not a Mersenne Prime.
    

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  • For p= 23 , 2²³ − 1 = 8388607 = 23 × 89 x 3 therefore 8388607 is not a Mersenne Prime.
    

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• M_n = 2ⁿ − 1 without the prime requirement may be called Mersenne numbers. And the smallest number is 2047 (2¹¹ – 1).
  

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• (2^p -1) is a Mersenne prime
  

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First nine Mersenne Primes and Perfect Numbers:

Rules  for perfect numbers by Nicomachus

• Nicomachus was a Pythagorean philosopher, and mathematician recognized as the father of Theoretic Arithmetic.
• Nicomachus had described certain results for perfect numbers (without proof)
  

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• • All perfect numbers are even.
  • All perfect numbers end in 6 and 8 alternately.
  • Euclid’s algorithm to generate perfect numbers:
  • N= 2^p-1(2^p – 1) ,where p is prime number & (2^p -1) is a Mersenne* prime
  • There are infinitely many perfect numbers