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# It’s abundant! Perfect! Prime!

## Every integer over 20,161 can be expressed as the sum of two abundant numbers

One thing I love about mathematics is the variety of names mathematicians have given to numbers. Here are a few, to give you an idea. Numbers can be natural, real, happy, perfect, square, rational, polygonal... you get the idea.

Also, numbers can be abundant. I don’t mean that there are a lot of them, which there certainly are. I mean there is a specific mathematical definition that classifies only certain numbers as “abundant". This is it: For any given integer, list its “proper divisors" - that is, its factors apart from the number itself. Add these factors. If the sum is greater than the original number, that original number is abundant.

As an example, take 10. Its factors are 1, 2 and 5, for a total of 8 - so 10 is not abundant. Maybe 11? It has just one proper divisor, 1. Not abundant either. Check 12: factors are 1, 2, 3, 4 and 6, totalling 16 - so 12 is abundant! In fact, it is the smallest abundant number. The next few are 18 (factors 1, 2, 3, 6, 9 for a total of 21), 20 (factors 1, 2, 4, 5, 10, total 22) and 24 (1, 2, 3, 4, 6, 8, 12, total 36).

Something may already have struck you about these first four abundants: they are all even. That’s one intriguing thing about these numbers. Make a list from the smallest one (12) on up and you could start to believe that abundants are always even. You’ll run through 100, then 150, then 200 abundants - in fact, 231 consecutive abundants before you hit the first odd one, 945.

This abundance of even abundants might have you wondering: is 945 the only odd abundant? Or are there just a few? Well, actually, there is an infinite number of abundant numbers, even and odd. One way you will believe this is if every multiple of an abundant is itself abundant - because obviously any number has an infinity of multiples. But is this so?

In fact, it is. Here’s an intuitive way to grasp this. Take 30, which is abundant because its proper divisors are 1, 2, 3, 5, 6, 10 and 15, adding to 42. Now, pick any multiple of 30 - 60, 270, 1410 - and call it M. Clearly M will have the above factors of 30 as its own factors, in addition to some more. Thus M/2, M/3, M/5, M/6, M/10 and M/15, whatever those numbers are, are also proper divisors of M. Totalling just these, we note first that

M/2 + M/3 + M/6 = M

And if we add M/5, M/10 and M/15 to this, we already have a number greater than M. Thus M is abundant.

So, if 945 is the first odd abundant, all its multiples are also abundant - and every alternate multiple is odd. So, even if we find no odd abundants other than 945 and its odd multiples, we know there’s an infinite number of them. (Though there are indeed odd abundants unconnected to 945.)

Take another nugget about abundants: every integer greater than 20,161 can be expressed as the sum of two abundant numbers. What’s special about 20,161, except that it’s prime? I don’t know as I write this, but I can’t wait to find out. But in playing with numbers this way, other links to primes also emerge. Wait for that.

Now, if we have abundants, we also have deficient numbers. Their principal divisors sum to less than the number itself. There’s 9, whose factors are 1 and 3, adding to 4. There’s 33, factors 1, 3, 11, for a total of 15. And of course, every single prime number is deficient. After all, each has only one principal divisor, 1. And as you can imagine, there are the occasional numbers whose factors sum to the number itself. Neither abundant nor deficient, these are called perfect. 6 is the smallest such, because its factors are 1, 2 and 3, which add to 6. That’s followed by 28,496 and 8128 and they quickly get much larger.

Ancient Greek mathematicians knew about perfect numbers. The great Euclid spelled out a relationship between the powers of 2 and perfect numbers that involves, yes, primes: what we now call Mersenne primes. These are prime numbers that are one less than a power of 2. If you have one such, said Euclid, multiply it by that power of 2 and divide by 2: you get a perfect number. (Try it with 32, the fifth power of 2 - you get 496.)

The largest primes we know of are Mersennes, and the continuing search for the next Mersenne is one of the great collaborative mathematical efforts of our time. As of May 2022, the largest of them all is 282589933 – 1, a monster that has 24,862,048 digits. Yes: That monster too produces a perfect number.

Yet, here are two simple things we don’t know about perfect numbers: one, are they infinite? And two, is there an odd perfect number? How odd that we know there are odd abundants and deficients, but we don’t know if there’s even one odd perfect! There are compensations: mathematical circles were abuzz recently, when a student at Oxford proved a conjecture that the stellar Paul Erdös made, involving primes and perfect numbers. That story, another time.

But it’s just more evidence of the endless connections you’ll find in the fabric of number theory. Abundant evidence, really.

Finally: Two thousand years after Euclid, the equally great German mathematician Leonhard Euler proved that all even perfect numbers have the particular link to Mersennes that Euclid found. That’s the Euclid-Euler theorem for you. That name is a tribute to two remarkable minds separated by two millennia, but a tribute too to the enduring charm and mystery of mathematics.

Once a computer scientist, Dilip D’Souza now lives in Mumbai and writes for his dinners. His Twitter handle is @DeathEndsFun.

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