Euclid's Derivation of the Perfect Number Formula
📂Number TheoryEuclid's Derivation of the Perfect Number Formula
If 2p−1 is a prime number, then 2p−1(2p−1) is a perfect number.
Explanation
It is not certain that all perfect numbers will have this form, but this form is definitely a perfect number.
For example, for the prime number (22−1)=3, 22−1(22−1)=6 is a perfect number.The fact that perfect numbers and Mersenne primes have this relationship could be somewhat inferred from the geometric series expansion of Mersenne primes.
Derivation
Since 2p−1 is a prime, the divisors of 2p−1(2p−1)
1,2,⋯,2p−1(2p−1),2(2p−1),⋯,2p−2(2p−1)
are divided into two categories. According to the sum of geometric series formula
1+2+⋯+2p−1=2−12p−1=2p−1
Similarly,
(2p−1)+2(2p−1)+⋯+2p−2(2p−1)=(2p−1−1)(2p−1)
Adding the two,
2p−1+(2p−1−1)(2p−1)=2p−1(2p−1)
Therefore, 2p−1(2p−1) is a perfect number.
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