Beyond Right and Wrong: Why Tackling Old Conjectures Makes You a Better Mathematician

In 1995, Andrew Wiles produced his proof of Fermat’s Last Theorem — an age old conjecture masquerading as a theorem (mostly because who would doubt Fermat when he writes that he had done the proof but it was too large for the margins?) and if you’ve ever had the audacity to approach a historically unproven postulate, you can imagine the grief Andrew Wiles had while working on this proof. Between the first conjecture in 1637 and Wiles’ proof in 1995, a long list of failed mathematics aimed to prove Fermat wrong. This isn’t inherently a bad thing — mathematics isn’t only about being right, it’s also about failing with grace and having the humility to learn from your mistakes.

When I started learning algebra as an undergraduate I began to realize how much of mathematics is balancing “Does it hurt to assume this?” with “Can we prove this?” I learned about this conjecture and that theory — like the Lonely Runner Conjecture and whether or not odd perfect numbers exist. Brocard’s Conjecture is particularly cool, I think. Maybe I just like Primes. One particular conjecture stands out to me, however, and is the focus of this article: The Carmichael Conjecture.

Nice Alliteration

The Carmichael Conjecture concerns itself with a property of the Euler Totient Function. The Euler Totient function has to do with identifying the number of elements in the multiplicative group of the integers mod n via the identification of numbers, k, which are less than the number n but greater than 1 which solve gcd(n, k) = 1. For a quick and dirty example, Euler’s totient function tells us that the cardinality of the multiplicative group of the integers mod 12 is the count of all numbers k such that 1 ≤ k < 12 and gcd(n, k) = 1. You can check this by hand, or you can use the Euler Totient Function’s multiplicity property and solve it really fast.

The Property:
Given a number n has prime factorization

the multiplicative property of the Euler Totient Function says

Where

ϕ(12) = ϕ(3)ϕ(2²) = [(3–1)*3⁰][(2–1)*2¹] = 2*2 = 4

and this is verified by the multiplicative group of the integers mod 12 being {1,5,7,11}

Now with that understanding of the multiplicity of the Euler Totient Function, consider the Carmichael Conjecture:

For Every n, ∃m : ϕ(n) = ϕ(m) where n ≠ m

We’re saying that for the multiplicative group of any given number, there exists another, different multiplicative group with the same cardinality. Seems simple enough, right? Here is the big punchline: this has only been proven for odd numbers! (see the proof below)

“So, why hasn’t it been solved?”

I spent the better part of my senior year and the summer after I graduated working on this question and I’ve made quite a bit of progress — it just seems so obvious, and it must be true! There are just so many numbers, how could there not be two numbers with the same totient function evaluation? But math doesn’t work on rhetorical “how could it not?” I was eager to talk to every one about it — between my esteemed professors at the University of Utah to my girlfriend to the geneticist working in a cancer lab with me. Each and everyone had the same kind of takeaway: “If it seems so obvious, why hasn’t it been solved?”

This is a fair question. I think most folks treat Mathematics as a right or wrong kind of science — no room for being wishy washy or unsure. This leads to a lot of issues when I can’t calculate the tip in my head in 2 seconds, or when I accidentally add two numbers together incorrectly in front of someone: people expect mathematics to be in a state of either proved or disproved — not acknowledging the legitimacy of the unproven, and when they do recognize the unproven they consider any still-to-come proofs futile to work on because “surely someone would’ve solved it by now!” Why spend time being wrong?

So here is my point: the time I spent working on this question — considered a futile, wasteful attempt by most — taught me to be patient, cautious and yet daring in my mathematics. It taught me style and finesse, and, most importantly, it allowed me to play. It showed me that it was okay to be wrong when people expected you to be wrong, and it was okay to take time to learn. It taught me what it means to be sure and that just because you can’t think of a counter example doesn’t mean there isn’t one. Don’t be afraid to get in over your head, and ask for help when you need it. Developing your mathematical maturity takes time, and it doesn’t come by always being right or doing the easy problems. Trust yourself enough to try and mathematics will be better because of it, and you will, too.

Odd Number Carmichael Proof

When a number n is odd, you can’t write it as n = 2m for some m. Knowing this, consider ϕ(2) = 1 and take n odd as 1, 3 mod 4