Perfect Numbers

Perfect Numbers - Even and Odd Perfect Numbers

Mathematics might seem like a dull and boring subject on the surface. We might think that nothing is happening except the simple algebraic calculations or the too difficult to comprehend, calculus. But the truth is, mathematics is a much more interesting subject contrary to the popular belief. Just like in any other subject, mathematics also deals with various branches of study and research. One of the branches pertaining to the theory, principles and basically the study of integers, is known as the number theory. As the great scientist, Carl Friedrich Guss said "mathematics is the queen of sciences and numbers is the queen of mathematics", numbers surely do dominate the field of this much-dreaded subject! Without numbers, there is no mathematics. There are some interesting numbers which are different from the usual integers that we use in our day to day calculations. In fact, we come across these numbers often but we are unable to spot the difference from the usual numbers.
One such type of number is called the perfect number. If we go by the modern-day definition of a perfect number, then we will define it by saying that, a perfect number is an integer, positive integers are considered, which will be equal to the summation of its proper divisors which are also to be considered positive. Proper divisors of a number are defined as the positive divisor of a number but except the divisor itself. For example, 1, 2, 4, 8, 16 are all divisors of the number 16 but when we have to write the proper positive divisors then we won’t take 16. In ancient times, perfect numbers were defined not in terms of divisors but in terms of aliquots. Aliquots are nothing but what we call the proper divisors or the proper quotients. 

The strange thing is that it is not known when the perfect numbers were first discovered! There are no records whatsoever which may indicate their time of discovery. The analysis of perfect numbers could have done in those ancient days when numbers first sparked a sudden interest in the mind of our ancestors. Nowadays scientists have started to believe that the existence of perfect numbers might have been around since the Egyptians, keeping in mind their usual way of calculating.
One more interesting fact about these special numbers is that they were studied by none other than Pythagoras himself along with his followers. In those days, magic and used to ignite much more curiosity than science and math. Same was the case with perfect numbers. They were studied not due to their specific pattern but due to their magical properties or so was thought. 

To get an idea of ancient these numbers are, let us look at the perfect numbers 6, 28, 496 and 8128. To check whether these really qualify as perfect numbers or not:

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

So, we can clearly see that the above- mentioned numbers are definitely a set of perfect numbers. Now to look at their history, we know that these numbers were first recorded in the book of Elements, written by Euclid in approximately 300 BC. In his book the Element, Euclid proved that if we take 2p−1 (2p − 1), then we will find that this is an even perfect number whenever 2p − 1 is taken as prime. 

In fact, the four perfect numbers that we saw before, are formulated on the basis of the above-mentioned formula only. We can see this by doing the following:

For the perfect number 6, if we put p=2 in the formula then we will get,

21 (22 − 1) = 2 × 3 = 6
Similarly, if we put p=3
22 (23 − 1) = 4 × 7 = 28
Putting p=5,
24 (25 − 1) = 16 × 31 = 496
Putting p=7,
26(27 − 1) = 64 × 127 = 8128.

Hence, we get these perfect numbers by keeping 2p − 1 as a prime number. The specialty about this formula is that it contains another type of special numbers called the Mersenne prime numbers. Mersenne prime numbers are those type of prime numbers whose value is always unity subtracted from a two’s power. 

We saw above the examples of even perfect numbers. So there exist even perfect numbers then is there something called as an odd perfect number as well? Whether odd perfect numbers are there or not is still not substantiated enough but there are various results or observations which have been made in the context of these odd perfect numbers. If we have to believe in what Jacques Lefèvre stated then there is no such thing as odd perfect numbers since all the perfect numbers that could be formulated were already done so using Euclid’s formula. In other words, all the perfect numbers have been formed already by Euclid. If we consider the recent theories then according to Carl Pomerance, the existence of odd perfect numbers is not possible. Also, there is more property that all perfect numbers should be able to fulfill. All perfect numbers should also be what is called an Ore's Harmonic Number. Now to define Ore's Harmonic Numbers, they are those numbers whose quotient or the divisors should have a mean integer value which should be harmonic in nature. Now, we know that only Ore's Harmonic number which is odd is 1. Therefore, the existence of any odd perfect number is not possible since no odd Ore's Harmonic number exists in mathematics apart from the number 1 thus, the reason behind Carl Pomerance’s theory. Let us assume that odd perfect numbers do exist, and then there are some criteria which they will have to fulfill in order to be a perfect number. Let us say the odd perfect number is X, then the criteria to be followed are:

  • X should not be divisible by 105.

  • X should be greater than 101500

  • X should be of the form:

  • X= qαp12e1 …....pk2ek

  • • Also, the X should have the largest prime factor as greater than 10^8.

  • • X should have a minimum of 101 prime factors and also a minimum of 10 different prime factors.