The theorem that made a mathematician
On the most recent episode of our My Favorite Theorem podcast, Kevin Knudson and I spoke with aBa Mbirika (who goes by the name aBa), a math professor at the University of Wisconsin Eau Claire. You can listen to the episode here or on kpknudson.comwhere there is also a transcript.
If you want to see aBa’s speech at the Joint Mathematics Meetings, which he mentioned in the episode, it was recorded, and you can watch it here.
As is the case with many of our guests, aBa struggled to choose a favorite theorem. He showed us around a few of his favorites before picking one out to tell us more about. The theorem on which he stopped is not an important theorem. You won’t find it in any manual. But it was important to aBa when he was a beginning math student.
Many people are familiar with the divisibility tests for 3 and 9: if the digits (in base ten) of a number add up to a multiple of 3, the number is divisible by 3. If they add up to a multiple of 9, the number is a multiple of 9. There are also easy ways to tell if a number is divisible by 2, 5, or 11. But you probably haven’t learned a divisibility test for 7. (There exists abut it’s not as easy to use as the rules for 3 and 9.)
The proof of divisibility by 9 uses modular arithmetic, where you only look at the remainders of numbers when they are divided by another number. It is also called clock arithmetic because we use mod 12 arithmetic when telling the time. (Five hours after 10:00 a.m. is 3:00 a.m. because 10 + 5 = 3 mod 12.) The proof that numbers are divisible by 9 if their digits add up to a multiple of 9 uses the fact that 10, the base of the system decimal number, is 1 mod 9. Divisibility rules use modular arithmetic because the basic question is: Does this number have a remainder of 0 when divided by another number?
The proof for 9 blew aBa’s mind, so soon after learning about it, he decided to see what else he could do with the general idea. On a long bus ride, he started with 7 to see if the same ideas could apply to a divisibility test for 7. Ten equals 3 mod 7, so the test he found uses many threes. In base ten, numbers are written as the sum of multiples of powers of ten. The number 3146 means 6×100+4×101+1×102+3×103. If we try to figure out what this mod number 7 is, we can change all these tens to three. Mode 7, 3846=6×30+4×31+1×32+3×33. So if this number is divisible by 7, so is 3846. So we can evaluate this expression: 6+12+9+81=108. And 108 isn’t a multiple of 7, so 3146 isn’t as good.
This rule for 7 does not scale as well as the rules for 3 and 9. The powers of 3 you need to calculate for rule 7 become unwieldy. Using this rule for a 10-digit number would require multiplying the leftmost digit by 59,049. the cyclic nature of the powers of 3 mod 7.) But even though the rule isn’t very helpful, aBa told us that understanding this theorem and its proof has changed his life. He created something new in math for the first time, and he was hooked.
Enrique Treviño, who was the guest of the Podcast Lathisms, told me a similar story. The problem that hooked him was to show that the sequence 2, 5, 8, 11, 14,…, where each successive number is 3 more than the last, has no perfect squares. The proof, which also uses modular arithmetic, is quite simple, but he describes it as an “eye-opening” experience that paved the way for his career in mathematics.
I suspect many other mathematicians have similar stories of simple, often unimpressive theorems and exercises that first made them feel they had the power to create in mathematics. I certainly felt that in my very first proofreading course. So today, I salute the tiny theorems that helped us become mathematicians.