Furthermore, the numbers can be abundant. I don’t mean that there are too many of them, that’s for sure. I mean there is a specific mathematical definition that classifies only certain numbers as “abundant”. That is: for a given integer, list its “proper divisor”—that is, its factors other than the number. Add up these factors. If the sum is greater than the original number, then that original number is abundant.
For example, take 10. Its factors are 1, 2 and 5, for a total of 8 – so 10 is not abundant. Maybe 11? It has only one proper denominator, 1. Not even abundant. Check 12: The factors are 1, 2, 3, 4 and 6, totaling 16 – so 12 is an abundance! 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, a total of 22) and 24 (1, 2, 3, 4, 6, 8, 12, 36 in total).
You may have already realized something about these first four abundances: they are all even. That’s an interesting thing about these numbers. Make a list from the smallest (12) above and you can begin to believe that the abundances are always the same. You’ll run into 100, then 150, then 200 in abundance – in fact, 231 in continuous abundance, before you hit the first odd number, 945.
Abundant Even Abundant You may wonder: is 945 the only odd abundant? Or are there only a few? Well, in fact, there are copious numbers, an infinite number of even and odd numbers. In a way you would believe that if each of the multiples of the multiplier is itself abundant – because clearly any number of multiples has an infinity. But is it so?
indeed it is. Here’s an intuitive way to understand it. Take 30, which is abundant because its proper divisors are 1, 2, 3, 5, 6, 10 and 15, which add up to 42. Now, choose any multiple of 30 – 60, 270, 1410 and call it M. Obviously the above factors of M 30 will be in addition to something else, as their own factors. Thus m/2, m/3, m/5, m/6, m/10 and m/15, whatever the numbers are, are also proper divisors of m. Taking these together, we first note 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 multiple, then all of its multiples are also abundant – and every alternate multiple is odd. So, even if we don’t find any odd multiples other than 945 and its odd multiples, we know there are an infinite number of them. (Although there are indeed strange copious amounts linked to 945.)
Take another nugget about abundance: Every integer greater than 20,161 can be expressed as the sum of two abundant numbers. What is special about 20,161 except that it is prime? I don’t know as I write this, but I can’t wait to find out. But playing with numbers in this way also reveals other links to prime numbers. wait for it.
Now, if we have copious numbers, we also have trivial numbers. Their prime divisors add up to less than the number itself. 9, whose factors are 1 and 3, which adds up to 4. is 33, factors 1, 3, 11, for a total of 15. And of course, every single prime number is trivial. After all, each has only one prime divisor, 1. And as you can imagine, there are sometimes numbers whose factors are sums of the numbers themselves. Neither abundant nor lacking, they are said to be perfect. 6 is the smallest because its factors are 1, 2 and 3, which add up to 6. Next come 28,496 and 8128 and they quickly become very large.
Ancient Greek mathematicians knew about whole numbers. The great Euclid explained a relationship between powers of 2 and the whole numbers, including, yes, primes: what we now call the Mersenne prime. These are prime numbers which are one less than the power of 2. If you have such one, Euclid said, multiply it by that power of 2 and divide by 2: you get a whole number. (Try it with 32, the fifth power of 2 – you get 496.)
The largest primes we know of are Mersennes, and the continued search for the next Mersenne is one of the great collaborative mathematical endeavors of our time. As of May 2022, the largest of them is 282589933 – 1, a monster with 24,862,048 points. Yes: that monster also produces a perfect number.
Still, there are two simple things we don’t know about whole numbers: One, are they infinite? And second, is there an odd whole number? How strange that we know that there is an odd abundance and a scarcity, but we don’t know if an odd is perfect! There are compensations: there was discussion in mathematical circles recently, when an Oxford student conjectured that the stellar Paul Erdos included prime numbers and whole numbers. That story, for the second time.
But this is just one more proof of the endless connections found in the fabric of number theory. Abundant evidence, indeed.
Finally: two thousand years after Euclid, the equally great German mathematician Leonhard Euler proved that all even whole numbers have a special relation to the mersens that Euclid found. That’s the Euclid-Euler theorem for you. The name is a tribute to two remarkable minds separated by two millennia, but also to the enduring fascination and mystery of mathematics.
Dilip D’Souza, once a computer scientist, now lives in Mumbai and writes for his dinner. His Twitter handle is @DeathEndsFun.