Morning Mathematical Monsters & Maniacs!
(Today’s post is sponsored by the letter “M”)
Over the past 600+ episodes, The Simpsons has taken us on an amazing mathematical journey involving fractions, probability, Fermat’s last theorem, and hundreds of other aspects from the wonderful world off mathematics.
And what better way to start your week, then by discussing math Monday morning?
So let’s get started this week by continuing our discussion from last week about Mersenne Prime numbers and move on to Perfect numbers.
In the finale of Season 17, Marge and Homer Turn a Couple Play (Season 17, Episode 22), the Springfield Isotopes’s star first baseman Buck “Home Run King” Mitchell is having marital problems with his pop star wife Tabitha Vixx (voiced by This Is Us star Mandy Moore). So when he sees Homer and Marge kissing on the Jumbotron, he asks them for advice. But when Homer is later caught giving a neck massage to Tabitha, both couples break up again – and it is hard for Homer and Marge to give advice when they are not talking to each other themselves.
Towards the end of the episode, Tabitha appears on the Jumbotron asking the stadium crowd to guess the evening’s attendance. At first glance, it appears as though the three numbers are just random numbers. However, as we saw last week, will see today and next week, these three numbers are more than just random numbers.
The second number on the Jumbotron is 8,128.
8,128 is a perfect number, which is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself (also known as its aliquot sum).
8,128 is the 4th Perfect number. The first 4 Perfect numbers are:
6 = 1 + 2 + 3
28 = 1 + 2 + 4 + 7 + 14
496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248
8,128 = 1 + 2+ 4 + 8 + 16 + 32 + 64 + 127 + 254 + 508 + 1,016 + 2,032 + 4,064
The first 4 Perfect numbers were discovered by the Ancient Greeks. The mathematician Nicomachus of Gerasa, born in the Roman province of Syria (modern day Jerash, Jordan) had noted 8,128 as early as 100 AD.
The giant of mathematics, Euclid of Alexandria, proved that 2n−1(2n − 1) is an even perfect number whenever 2n − 1 is prime (to be specific, a Mersenne prime as discussed last week).
For example, the first four perfect numbers are generated by the formula 2n−1(2n − 1), with n a prime number, as follows:
- for n = 2: 21(22 − 1) = 2 × 3 = 6
- for n = 3: 22(23 − 1) = 4 × 7 = 28
- for n = 5: 24(25 − 1) = 16 × 31 = 496
- for n = 7: 26(27 − 1) = 64 × 127 = 8128
As you may recall from last week, 3, 7, 31, and 127 are the first four Mersenne primes.
Now that we know the secret behind the second number on the Jumbotron, be sure to come back next week when we talk about the third and final number.
Do you remember this episode? Did you catch the number 8,128 on the Jumbotron? Did you recognize the number 8,128 as a perfect number? Are you familiar with Perfect numbers? Sound off in the comments below. You know we love hearing from you.