Mystery of ‘Perfect Numbers’ Resolved – Perfect Number Is Always Even and Predictable

Abstract

In number theory, a perfect number is a positive integer that is equal to the sum of its proper positive divisors, excluding the number itself. In other words, a perfect number is a number that is half the sum of all of its positive divisors (including itself) i.e. σ1(n) = 2n. To explain in practical terms, we elaborate first few Perfect Numbers. It may be noted that ‘Perfect Numbers’ are sparse are thinly dispersed. Starting from 3rd Century BC, mathematicians are working on Perfect Numbers. Till April 2018, i.e. during last 2300 years active research, researchers could recognize only 50 perfect numbers. There are 2 perfect numbers in first 100 and 4 in first million. Absolute distance between two perfect numbers increase exponentially as you go higher to the next perfect number . One can find at least one perfect number till 4 digit numbers, and then it becomes a real rarity. Subsequent perfect numbers appears at 8, 10, 12 and 19 digits. 15th perfect number has 770 digits while 16th have 1327 digits. 25th perfect number has 13066 digits. 50th perfect number has 46,498,850 digits. We found that perfect number is always predictable by using formula 1 (p) 0 (p-1) where 1 and 0 are binary digits and p = count of binary digit. We also argue that if any binary number 1...(p) 0 (p-1) if perfect number, will always an even number. We also observed that first known 50 perfect number ends with 6 or 28 as last one or two digits. Therefore a perfect number is always predictable and even.