Abundant, Deficient, and Perfect Numbers

Numbers can be classified in many ways. They are either even or odd, prime or composite (or 1), they may have more or less than 12 divisors, etc. A classification that was introduced by the ancient Greeks (who ascribed mystical properties to numbers), and that has proved interesting in more modern studies of number theory from a mathematical perspective, involves the sum of the divisors of a number. The Greeks thought about divisors of a number as the "parts" of the number, and so they did not consider the number to be a part of itself. So we need one more new idea (not a toughie!):

Proper divisors: A number x is a proper divisor of y provided that x is a divisor of y and x is less than y.

So the only "improper" divisor of a number is the number itself; all other divisors are proper. The Greeks classified numbers according to the sums of their proper divisors.

We'll write P(n) for the sum of all the proper divisors of the number n.

Example: P(15)=9, because the proper divisors of 15 are 1, 3, and 5, and 1 + 3 + 5 = 9.

The problem with the sum of proper divisors is that P(n) does not have the nice multipliciative property that S(n), the sum of all the divisors, had. But we can still compute P(n) pretty easily, using the formula for S(n) -- since the only difference is that we don't add the n in P(n). Therefore:

Example: Let's compute P(1800). To do this, first we'll compute S(1800) and then subtract off 1800. Start with the prime factorization 1800 = 2^{3 .} 3^{2 .} 5². Then calculate:

1 + 2 + 2² + 2³ = 15
1 + 3 + 3² = 13, and
1 + 5 + 5² = 31.

Therefore S(1800) = (15)(13)(31) = 6045. This is the sum of all the divisors of 1800 (including 1800)itself. To get P(1800), we just subtract off the 1800 and get that

P(1800) = S(1800) - 1800 = 6045 - 1800 = 4245.

The Greeks observed an interesting distinction, that you can observe in the two examples above. First, it's not at all surprising that P(1800) is much larger than P(15), but it's noteworthy that P(1800) is in fact larger than 1800, whereas P(15) is smaller than 15. This leads to the Greek classification of numbers as follows:

• If P(n) > n, then n is called an abundant number.
• If P(n) < n, then n is called a deficient number.
• If P(n) = n, then n is called a perfect number.

A couple of problems for you:

1. Classify the numbers 28, 50, 120 and 4173 as being abundant, deficient, or perfect. You might find the Prime Factorization Calculator helpful.
2. Are prime numbers abundant, deficient or perfect?
3. Show that every multiple of 12 is abundant. (Hint: A multiple of 12 can be written as 12n for some number n. Find enough proper divisors of 12n to add up to more than 12n. It's ok if you don't find them all, since adding the ones you don't find can only make P(12n) bigger.) This shows there are infinitely many abundant numbers.

And next week we'll zero in on the perfect numbers.