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?
This week, we’re going to step away from the TV show and talk about a mathematical number relevant to the game.
As many of you have often asked, there is an upper limit to how much in-game cash you can accumulate. That limit is $4,294,967,295.
Once you reach $4,294,967,295, some fun dialogue pops up between Prof. Frink and Homer and the in-game cash turns red. It does not increase beyond this point. But why is this number significant?
4,294,967,295 is equal to 232 − 1. As you may recall from three weeks ago when we discussed Mersenne Primes, Mersenne Primes also come in the form 2n − 1. However, 4,294,967,295 is not a Mersenne Prime, as it has a factorization of 3 x 5 x 17 x 257 x 65,537.
These 5 factors may look familiar to some of you, as these are the 5 Fermat prime numbers. Fermat primes were first studies by the French giant of mathematics Pierre de Fermat (pictured below).
Fermat believed that F(n) = 22^n + 1 were prime numbers. He proved that the first 5 such numbers were indeed prime:
F(0) = 22^0 + 1 = 21 + 1 = 2 + 1 = 3
F(1) = 22^1 + 1 = 22 + 1 = 4 + 1 = 5
F(2) = 22^2 + 1 = 24 + 1 = 16 + 1 = 17
F(3) = 22^3 + 1 = 28 + 1 = 256 + 1 = 257
F(4) = 22^4 + 1 = 216 + 1 = 65,536 + 1 = 65,537
In 1732, Swiss mathematician Leonhard Euler proved that F(5) is not a prime number. Over the past 300 years, several other Fermat numbers have been shown not to be primes. To date, only these first 5 numbers proved by Fermat have been shown to be Fermat primes.
Aside from being factorized by the 5 Fermat primes, 4,294,967,295 is important in the world of computing. The number 4,294,967,295, equivalent to the hexadecimal value FFFF,FFFF 16; which is the maximum value for a 32-bit unsigned integer in computing.
This value is the largest memory address for CPUs using a 32-bit address bus. It is therefore the maximum value for a variable declared as an unsigned integer in many programming languages running on modern computers.
Now we know why $4,294,967,295 is the limit of how much in-game cash you can earn and the significance of the number in both the fields of mathematics and computing.
Have you reached $4,294,967,295 in-game cash? Did you know the significance of the number? Are you familiar with Fermat primes? Sound off in the comments below. You know we love hearing from you.