Perfect numbers like perfect men are very rare.
– Rene Descartes
After having a look at applying mathematics, I thought before elaborating my last post why not have a dig at how maths can be absolute fun. Perfect numbers to the best of my knowledge don’t find any direct application in real world. If you know any please mention it in comments (I’ll be the most happy person if there is one 🙂 ) .
Before starting let’s define some terms to make things easy;
Aliquot sum: sum of all divisors of an integer excluding itself. e.g. for 12 it is 1+2+3+4+6 = 16. Sometimes it is also denoted as s(n), the restricted divisor function.
Divisor function: sum of all divisors of an integer including itself. Denoted as σ(n).
Perfect number: A positive integer which is equal to its aliquot sum, e.g. 6 = 1+2+3.
Greeks knew first 4 – 6,28,496,8128.
One thing which mathematician always does is tries to formulate. So is there a formula to calculate next perfect number? Euclid proved that 2p−1(2p−1) is an even perfect number whenever 2p−1 is prime. Euler further proved that this formula will yield all even perfect numbers. We are not going into Euler’s proof a more general one but we will see Euclid’s proposition.
A quick note would be any even number can be written as m.2n-1, with m,n ε Z and m,n > 0.
Lets start with a perfect number p = q.2r-1 ,with q being prime. As p is perfect σ(p) = 2p. Moving on as q is prime σ(q) = q+1. Also σ(2r-1) = 2r – 1 # I don’t think it needs any explanation.
Now, let’s combine one more identity with them.
σ(p1α1 p2α2… prαr) = σ(p1α1)σ( p2α2)…σ( prαr) where n = p1α1 p2α2… prαr is prime factorization of n.
Thus, σ(p) = σ(q).σ(2r-1) => σ(p) = (q+1)(2r -1). As p is perfect by assumption LHS equals 2p = 2q.2r-1
Hence simplifying we get q = 2r -1, thus Euclid’s form obtained.
One can also note that primes of the form 2r -1 make a special class of primes called Mersenne primes. Thus perfect numbers and mersenne primes are intimately connected. Owing to their form all even perfect numbers in binary are n ones followed by n-1 zeroes with n being prime.
610 = 1102 , 2810 = 111002 , 49610 = 1111100002 ,812810 = 11111110000002
So what about odd ones? Well it is unknown whether they exist at all. The recently published result shows if at all they do they must be > 101500 .
I am not going to discuss about odd perfects as I myself is not very much aware about them. Though one interesting result I recently got to know is no odd perfect number is divisible by 105.
– Reciprocals of divisors add up to 2. e.g. for 6 : 1/1+1/2+1/3 + 1/6 = 2. (easy proof σ(n) = 2n. Divide both sides by n to get the result)
– A perfect number is never a perfect square. Thus it always has even no. of divisors. This deduces
– An even perfect no. is always a Ore’s Harmonic number. i.e. Divisors have an integral harmonic mean. (Again easy to prove; reciprocals add to 2 and no. is even) And one last
– Even perfect nos. are not trapezoidal. i.e. They can’t be represented as sum of 2 or more consecutive positive integers or in other words difference of 2 triangular nos. #Triangular no. = 0.5n(n+1)
Interestingly there are only 3 such categories. Even perfects, powers of 2 and a class of nos. formed by Fermat primes similar to how even perfects are formed from Mersenne primes.
There are generalizations of perfect nos. like superperfect nos.
A no. n is called superperfect if σ2(n) = σ(σ(n)) = 2n. e.g. 2,4,16,64,4096,65536 are first few superperfects.
Even more generalization is (m,k) perfect numbers defined as – σm(n) = kn with m,k being integers
e.g. 12 is a (3,any) perfect number. #check yourself. It will stand for any integral k.
There are many classes of perfect nos. Numerous specializations and generalizations but still without any application in real world to the best of my knowledge. Then why they are here? Let’s see what great mathematician J.E. Littlewood says
The theory of numbers is particularly liable to the accusation that some of its problems are the wrong sort of questions to ask. I do not myself think the danger is serious; either a reasonable amount of concentration leads to new ideas or methods of obvious interest, or else one just leaves the problem alone. “Perfect numbers” certainly never did any good, but then they never did any particular harm.
Still not getting it? They are for fun 😀
 Odd perfect numbers – Ochem, P; Rao, M (2012).