Perfection in numbers – 03/26/2024 – Marcelo Viana

“Professor, your course started on January 6th and will end on February 28th. Now 6 is a perfect number and so is 28! What does this prove about the course?” I heroically resisted replying that my course was perfect, and moved on to talk about the mathematics behind the student observation.

The notion is very old: a number is said to be perfect if it is equal to the sum of its own divisors, that is, the divisors that are smaller than the number itself. For example, 6 is perfect because its proper divisors are 1, 2 and 3 and their sum gives precisely 6. Similarly, the proper divisors of the number 28 are 1, 2, 4, 7 and 14, and their sum is equal to 28.

The ancient Greeks knew the first four perfect numbers: after 6 and 28 come 496 and 8128. They have intriguing properties. For example, they are all triangular numbers, that is, they are sums of the first integers: 6=1+2+3, 28=1+2+3+4+5+6+7, 496=1+2+3+ …+30+31 and 8128=1+2+3+…+126+127.

The first important result about perfect numbers was proved by Euclid, around 300 BC: he showed that if p is a prime number such that 2^P-1 is also prime so N=2^P-1(two^P-1) is a perfect number. For p=2 this gives N=6, for p=3 it gives N=28, for p=5 it gives N=496, and for p=7 it gives N=8128.

From there it gets more complicated. The next candidate would be p=11, but 2¹¹-1=2047 is not prime, and the corresponding N=2,096,128 is not perfect. In fact, the next perfect numbers would take a millennium and a half to be discovered.

Around 1230, the Egyptian mathematician Ibn Fallus (1194–1252) published a list that he said contained ten perfect numbers. In fact, three were wrong, but he still added three perfect (correct) numbers to the four known to the Greeks: 33,550,336, 8,589,869,056, and 137,438,691,328. However, his work was not published in Europe, and ended up being rediscovered in the Italian Renaissance: 33,550,336 appeared in an anonymous manuscript around 1456, and the other two numbers were exhibited by the Bolognese Pietro Cataldi (1548–1626) in 1588.

All of these discoveries were based on the formula N=2^P-1(two^P-1) by Euclid, and then the obvious question arose: aren’t there other ways to find perfect numbers? In particular, noting that Euclid’s formula only gives even numbers, do there not also exist odd perfect numbers?

None other than the great Leonhard Euler enters the scene.

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