Number Perfection

I’ve been waiting to make this one for a while now.

My birthday is particularly special to me. I know that more and more people don’t treat their birthdays as seriously as others (one of my closest friends from college generally disregards his), I for one have always enjoyed mine, and it all has to do with the numbers in them.

Let us discuss my actual birthday itself: June 28. There are quite a few well-known folks who share a birthday with me: Mel Brooks, Kathy Bates, Seo Joo-hyun (better known as Seohyun), and Bradley Beal, to name a few. But why June 28? What makes it so special?

To begin, let us discuss factors, a subject that most often comes up in elementary school. A factor of a number “n” is a number that evenly divides “n”. This notion is why the term “divisor” is used in higher education and academia, as it plays into how division works: there’s a dividend, a divisor, and a quotient, and given a dividend, its divisors leave a quotient with no remainder when division is performed. For example, with 12, the factors are 1, 2, 3, 4, 6, and 12.

Now we have briefly touched on factors before, specifically with prime numbers. Prime numbers have exactly two factors: one and itself. Note that 1 does not count as prime because it has only one, not two. Composite numbers have three or more, and 12 is a good example as it has six. Most composite numbers will have an even number of factors thanks to each factor having another factor to pair with that multiply to the original number. As a consequence, square numbers have an odd number of factors thanks to its square root “pairing” with itself.

So beyond prime numbers, are there other cool things involving factors? Sure! There’s Greatest Common Factor of two numbers and its relation to the Least Common Multiple of two numbers (namely that given two numbers “m” and “n”, the product of their GCF and LCM is just m x n), but possibly the most intriguing is that of the “aliquot sum”, a number that is used in elementary school but is not named. The aliquot sum of a number “n” is merely the sum of the proper factors of “n”, where a proper factor is merely any factor that isn’t “n” itself.

Using the earlier example of 12, the proper factors are 1, 2, 3, 4, and 6, and the aliquot sum is 16. Why not try it with a number like 15? Well our proper factors here are 1, 3, and 5, meaning the aliquot sum is 9. Notice how with 12, the aliquot sum is greater than 12, yet with 15, the aliquot sum is smaller. The relation between the aliquot sum and its relation to the original number allows us to assign a designation. Should the aliquot sum be greater, the number is said to be “abundant”. If it is smaller, the number is said to be “deficient”.

For example, all prime numbers are deficient as they have just one proper factor, 1, meaning their aliquot sum is 1 which is always less than the original number. Square numbers see a mix of both abundant and deficient. 256, which is 16², is deficient as its aliquot sum is 255, yet 144, which is 12², is abundant as its aliquot sum is 259.

Now this begs the question that’s been lying out there: are there numbers where the aliquot sum is EQUAL to the original number? The answer is yes, and these numbers are given the moniker of “perfect”. The smallest such number is 6, as the proper factors are 1, 2, and 3, meaning the aliquot sum is 6. The next perfect number is 28, as the proper factors are 1, 2, 4, 7, and 14, and the aliquot sum is 28. The next two perfect numbers are 496 and 8128. Now notice how the first perfect number is one digit, the second has two, the third has three, and the fourth has four. Does the fifth have five? No, as the next one is 33550336. Yikes.

There are important things that came about perfect numbers. Euclid proved that if 2^p – 1 is prime where p is prime, then 2^p-1(2^p – 1) is an even perfect number. For p = 2, we get 6, and then for p = 3, we get 28. This follows for p = 5 (which gives 496) and p = 7 (which gives 8128), but NOT p = 11, as 2¹¹ – 1 = 2047 = 23 x 89. Such prime numbers that allow 2^p – 1 to be prime are referred to as Mersenne primes, named after Marin Mersenne. Euler would later go on to prove that all even perfect numbers follow the form that Euclid gave from his theorem (effectively proving the converse of Euclid’s theorem).

Now you might be wondering: we’ve talked about even perfect numbers, what about odd ones? Well as it turns out, no one has any clue if there exist any. Such a simple question, yet proving their existence or non-existence has been difficult for even the best intellects. Even Euler himself said, “Whether there are any odd perfect numbers is a most difficult question.”

One question I asked myself was whether or not any odd abundant numbers exist, and that should none exist, perhaps that could have been used to show how no odd perfect number exists. However, based on my own research with repeating decimals (more on that in a later post) as well as other reading, I found that odd abundant numbers DO exist, and that the lowest one is 945.

There are more interesting numbers related to abundant, deficient, and perfect numbers, but that’s a subject that I’m not well-versed in at all, although once I come to learn more about them, I’ll talk about my findings. Until then, enjoy your day, and happy birthday to me, I guess.