logo

Euclid's Derivation of the Perfect Number Formula 📂Number Theory

Euclid's Derivation of the Perfect Number Formula

Formula 1

If 2p12^{p}-1 is a prime number, then 2p1(2p1)2^{p-1}(2^{p} - 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 (221)=3(2^2 -1) = 3, 221(221)=62^{2-1}(2^2 -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 2p12^{p}-1 is a prime, the divisors of 2p1(2p1)2^{p-1}(2^{p} - 1) 1,2,,2p1(2p1),2(2p1),,2p2(2p1) 1,2, \cdots , 2^{p-1} \\ (2^{p}-1), 2 (2^{p}-1), \cdots , 2^{p-2} (2^{p}-1) are divided into two categories. According to the sum of geometric series formula 1+2++2p1=2p121=2p1 1 + 2 + \cdots + 2^{p-1} = { {2^{p}-1} \over {2 - 1} } = 2^{p}-1 Similarly, (2p1)+2(2p1)++2p2(2p1)=(2p11)(2p1) (2^{p}-1) + 2 (2^{p}-1) + \cdots + 2^{p-2} (2^{p}-1) = ( 2^{p-1} - 1 ) (2^{p}-1) Adding the two, 2p1+(2p11)(2p1)=2p1(2p1) 2^{p}-1 + ( 2^{p-1} - 1 ) (2^{p}-1) = 2^{p-1}(2^{p} - 1) Therefore, 2p1(2p1)2^{p-1}(2^{p} - 1) is a perfect number.


  1. Silverman. (2012). A Friendly Introduction to Number Theory (4th Edition): p102. ↩︎