Episode 50 - aBa

My Favorite Theorem

Evelyn Lamb: Hello, and welcome to My Favorite Theorem, a math podcast. I'm one of your hosts, Evelyn Lamb. I'm a freelance math and science writer, usually in Salt Lake City, Utah, currently in Providence, Rhode Island. And this is your other host.
Kevin Knudson: Hi. I’m Kevin Knudson, professor of mathematics, almost always at the University of Florida these days. How's it going?
EL: All right. We had hours of torrential rain last night, which is something that just doesn't happen a whole lot in Utah but happens a little more often in Providence. So I got to go to sleep listening to that, which always feels so cozy, to be inside when it's pouring outside.
KK: Yeah, well, it's actually finally pleasant in Florida. Really very nice today and the sun's out, although it's gotten chilly—people can't see me doing the air quotes—it’s gotten “chilly.” So the bugs are trying to come into the house. So the other night we were sitting there watching something on Netflix and my wife feels this little tickle on her leg and it was one of those big flying, you know, Florida roaches that we have here.
EL: Ooh
KK: And our dog just stood there wagging at her like, “This is fun.” You know?
EL: A new friend!
KK: “Why did you scream?”
EL: Yeah, well, we’re happy today to invite aBa to the show. ABa, would you like to introduce yourself?
aBa Mbirika: Oh, hello. I’m aBa. I'm here in Wisconsin at the University of Wisconsin Eau Claire. And I have been here teaching now for six years. I tell them where I'm from?
EL: Yeah.
KK: Sure.
aM: Okay. I am from, I was born and raised in New York City. I prefer never to go back there. And then I moved to San Francisco, lived there for a while. Prefer never to go back there. And then I went up to Sonoma County to do some college and then moved to Iowa, and Iowa is really what I call home. I'm not a city guy anymore. Like Iowa is definitely my home.
EL: Okay.
KK: So Southwestern Wisconsin is also okay?
aM: Yeah, it's very relaxing. I feel like I'm in a very small town. I just ride my bicycle. I still don't know how to drive, like all my friends from New York and San Francisco. But I don't need a car here. There's nowhere to go.
EL: Yeah.
aM: But can I address why you just called me aBa, as I asked you to?
EL: Yeah.
aM: Yeah, because maybe I'll just put this on the record. I mean, I don't use my last name. I think the last time I actually said some version of my last name was grad school, maybe? The year 2008 or something, like 10 years ago was the last time anyone's ever heard it said. And part of the issue is that it's It's pronounced different depending on who's saying it in my family. And actually it's spelled different depending on who’s in the family. Sometimes they have different letters. Sometimes there's no R. Sometimes it’s—so in any case, if I start to say one pronunciation, I know Americans are going to go to town and say this is the pronunciation. And that's not the case. I can't ask my dad. He's passed now, but he didn’t have a favorite. He said it five different ways my whole life, depending on context. So he doesn't have a preference, and I'm not going to impose one. So I'm just aBa, and I'm okay with that.
EL: Yeah, well, and as far as I know, you're currently the only mathematician named aBa. Or at least spelled the way yours is spelled.
aM: Oh yeah, in the arXiv. Yeah, on Mathscinet that it’s. Yeah, I'm the only one there. Recently someone invited me to a wedding and they were like, what's your address? And I said, “aBa and my address is definitely enough.”
EL: Yeah, so what theorem would you like to tell us about?
aM: Oh, okay, well I was listening actually to a couple of you shows recently, and Holly didn’t have a favorite theorem, Holly Krieger. I'm exactly the same way. I don't even have a theorem of, like, the week. She was lucky to have that. I have a theorem of the moment. I would like to talk about something I discovered when I was in college, that’s kind of the reason. but can I briefly say some of my like, top hits just because?
EL: Oh yeah.
KK: We love top 10 lists. Yeah, please.
aM: Okay. So I'm in combinatorics, loosely defined, but I have no reason—I don't know why people throw me in that bubble. But that's the bubble that that I've been thrown in. But my thesis—actually, I don’t ever remember the title, so I have to read it off a piece of paper—Analysis of symmetric function ideals towards a combinatorial description of the cohomology ring of Hessenberg varieties.
KK: Okay.
aM: Okay, all those words are necessary there. But my advisor said, “You're in combinatorics.” Essentially, my problem was, we were studying an object and algebraic geometry, this thing called a Hessenberg variety. To study this thing we used topology. We looked at the cohomology ring of this, but that was very difficult. So we looked at this graded ring from the lens of commutative algebra. And I studied the algebra the string by looking at symmetric functions, ideals of symmetric functions, and hence that's where my advisor said, “You're in combinatorics.” So it was the main tool used to study a problem an algebraic geometry that we looked at topology. Whatever, so I don't know what I am. But any case for top 10 hits, not top 10, but diagram chasing. Love it. Love it.
EL: Wow, I really don't share that love, but I’m glad somebody does love it.
aM: Oh, it's just so fun for students.
KK: So the snake lemma, right?
aM: The snake lemma, yes. It's a little bit maybe above the level of our algebra two class that I teach here for undergrads, but of course I snuck it in anyways. And the short five lemma. Those are like, would be my favorites if the moment was, like, months ago. In number theory I have too many faves, but I’m going to limit it to Euler-Fermat’s theorem that if a and n are coprime, then a to the power of the Euler totient function of n is congruent to 1 mod n. But that leads to Gauss’s epically cool awesome theorem on the existence of primitive roots. Now, this is my current craze.
EL: Okay.
aM: And this is just looking at the group of units in Z mod nZ, or more simply the multiplicative group of units of integers modulo n. When is this group cyclic? And Gauss said it's only cyclic when n is the numbers 2, or 4, or an odd prime to a k power, or twice an odd prime to some k-th power. And basically, those are very few. I mean, those are very little numbers in the broad spectrum of the infinity of the natural numbers. So this is very cool. In fact, I'm doing a non-class right now with a professor who retired maybe 10 years ago from our university, and I emailed him and said, “Want to have fun on my like my research day off?” And we’re studying primitive roots because I don't know anything about it. Like, my favorite things are things I know nothing about and I want to learn a lot about.
EL: Yeah, I don't think I've heard that theorem before. So yeah, I'll have to look that up later.
aM: Yes. And then the last one is from analysis, and I did hear Adrianna Salerno talked about it and in fact, I think also someone before her on your podcast, but Cantor’s theorem on uncountability of the real numbers.
EL: Yeah, that's that's a real classic.
aM: I just taught that two days ago in analysis, and like, it's like waiting for their heads to explode. And I think, I don't know, my students’ heads weren't all exploding. But I was like, “This is so exciting! Why are you not feeling the excitement?” So yeah, yeah, it was only my second time teaching analysis. So maybe I have to work on my sell.
EL: Yeah, you'll get them next time.
aM: Yeah. It's so cool! I even mentioned it to my class that’s non-math majors, just looking at sets, basic set theory. And this is my non-math class. These students hate math. They're scared of math. And I say, “You know, the infinity you know, it's kind of small. I mean, you're not going to be tested on this ever. But can I please take five minutes to like, share something wonderful?” So I gave them the baby version of Cantor’s theorem. Yeah, but that's it. I just want to throw those out there before I was forced to give you my favorite theorem.
EL: Yes. So now…
KK: We are going to force you, aBa. What is your favorite theorem?
EL: We had the prelude, so now this is the main event.
aM: Okay, main event time. Okay, you were all young once, and you remember—oh, we’re all young, all the time, sorry—but divisibility by 9. I guess when we're in high school—maybe even before that—we know that the number 108 is divisible by 9 because 1+0+8 is equal to 9. And that's divisible by 9. And 81 is divisible by 9 because 8+1 is 9, and 9 is divisible by 9. But not just that, the number 1818 is divisible by 9 because 1+8+1+8 is 18. And that's divisible by 9. So when we add up the digits of a number, and if that sum is divisible by 9, then the number itself is divisible by 9. And students know this. I mean, everyone kind of knows that this is true. I guess I was a sophomore in college. That was maybe a good 4 to 6 years after I started college because, well, that was hard. It's a different podcast altogether, but I made some choices to meet friends who made it really hard for me to go to school consistently in San Francisco—part of the reason why I'm kind of okay not going back there much anymore. Friends got into trouble too much.
But I took a number theory course and learned a proof for that. And the proof just blew my mind because it was very simple. And I wasn't a full-blown math major yet. I think I was in physics— I had eight majors, different majors through the time—I wasn't a math person yet. And I was on a bus going from—Oh, this is in Sonoma County. I went to Sonoma State University as my fourth or fifth college that I was trying to have a stable environment in. And this one worked. I graduated from there in 2004. It definitely worked. So I was on a bus to visit some of my bad friends in San Francisco—who I love, by the way, I'm just saying of the bad habits—and I was thinking about this theorem of divisibility by 9 and saying, what about divisibility by 7? No one talks about that. Like, we had learned divisibility by 11. Like the alternating sum of the digits, if that's divisible by 11, then the number is divisible by 11. But what about 7? You know, is that doable? Or why is it not talked about?
EL: Yeah.
aM: So it was an hour and a half bus ride. And I figured it out. And it was extremely, like, the same exact proof as the divisibility by 9, but boiled down to one tiny little change. But it's not so much that I love this theorem. I actually haven't even told it to you yet. But that I did the proof, that it changed my life. I really—that’s the only thing I can go back to and say why am I an associate professor at a university in Wisconsin right now. It was the life-changing event. So let me tell you the theorem.
EL: Yeah.
aM: It’s hardly a theorem, and this is why I don't know if it even belongs on this show.
EL: Oh, it totally does!
aM: Okay, so I don't even think I had calc 2 yet when I discovered this little theorem. All right, so here we go. So look at the decimal representation of some natural number. Call it n.
EL: I’ve got my pencil out. I'm writing this down.
aM: Oh, okay. Oh, great. Okay, I'm reading off a piece of paper that I wrote down.
EL: Yeah, you said something about it to us earlier. And I was like, “I'm going to need to have this written down.” It’s funny that I do a podcast because I really like looking at things that are written down. That helps me a lot. But let's podcast this thing.
aM: Okay, so say we have a number with k+1 digits. And so I'm saying k+1 because I want to enumerate the digits as follows: the units digit I'm going to call a0, the tens digit I’ll call a1 the hundreds place digit a2 etc, etc, down to the k+1st digit, which we’ll call ak. So read right to left, like in Hebrew, a0, a1 a2 … (or \cdots, you LaTeX people) ak-1 then the last far left digit ak.
EL: Yeah.
aM: So that is a decimal representation of a number. I mean, we're just, you know, like number 1008. That would be a0 is the number 8, a1 is the number 0, a2 is number 0, a3 is the number 1. So we just read right to left. So we can represent this number, and everybody knows this when you're in junior math, I guess in elementary school, that we can write the number—now I'm using a pen—123 as 3 times 1 plus—how many tens do we have? Well, we have two tens. So 2 times 10. How many hundreds do we have? Well, we have one of those. So 1 times 100. So just talking about, yeah, this is mathematics of the place value system in base 10. No surprise here. But a nicer way to write it as a fat sum, where i, the index goes from 0 to k of ai times 10i.
EL: Yeah.
aM: That’s how we in our little family of math nerds, how we compactly write that. So when we think about when does this number divisible by 7? It suffices to think about when what is the remainder when each of these summands is—when we divide each of these summands by 7, and then add up all those remainders and then take that modulo 7. So the key and crux of this argument is that what is 10 congruent to mod 7? Well, 10 leaves the remainder of 3 when you divide by 7. In the great language of concurrences—Thank you, Gauss—10≡3 mod 7. So now we can look at this, all of these tens we have. We have a0 ×100+ a1 ×101 + a2 ×102, etc, etc. When we divide this by 7, this number really is now a0 ×30 — because I can replace my 100 with 30 —plus a1 ×31 instead of—because 101 is the same as 31 in modulo 7 land—plus a2 ×32, etc. etc…. to the last one, ak ×3k. Okay, here I am on the bus thinking, “This is only cool if I know all my powers of 3.”
EL: Yeah. Which are not really that much easier than figuring it out in the first place.
aM: Okay, but I'm young mathematically and I'm just really super excited. So one little example, I guess this is not, I can't remember what I did on the bus, but 1008 is is a number that's divisible by seven. And let's just perform this check, using this check on this number. So is 1008 really divisible by 7? What we can do is according to this, I take the far right digit, the units digit, and that's 8 ×30, so that's just the number 8, 8×1, plus 0×31. Well, that's just 0, thankfully. Then the next, the hundreds place, that’s 0×32. So that's just another 0. And then lastly, the thousands place, 1×33 and that's 27. Add up now my numbers 8+0+0+27. And that's 35. And that's easy to know that the divisibility of. 7 divides 35 and thus 7 divides 1008. And, yeah, I don't know, I’m traveling back in time, and this is not a marvelous thing. But everybody, unfortunately, who I saw in San Francisco that day, and the next day, learned this. I just had to teach all my friends because I was like, “Well, this is not what I'm doing for college. This is something I figured out on the bus. This math stuff is great.”
EL: Yeah, just the fact that you got to own that.
aM: Yeah. And that also it wasn't in the book, and actually it wasn't in subsequently any book I've ever looked in ever since. But it's still just cute. I mean, it's available. And what it did, I guess it just touched me in a way, where I guess I didn't know about research, I didn't know about a PhD program. My end goal was to get a job, continue at the photocopy place that was near the college, where I worked. I really told my boss that, and I really believed that I was going to do that. And our school never really sent people to graduate programs. I was one of the first. And I don't know, it just changed me. And there were a lot of troubles in my life before then. And this is something that I owned. And that's my favorite theorem on that bus that day.
KK: It’s kind of an origin story, right?
aM: Yes, because people ask me, how did you get interested in math? And I always say the classic thing. Forget this story, but I'm also not speaking to math people. My usual thing is the rave scene. I mean, that was what I was involved in in San Francisco, and then, I don't know if you know what that is, but electronic dance music parties that happen in beaches and fields and farms and houses.
EL: What, you don’t think we go to a lot of raves?
aM: I don’t know if raves still happen!
EL: You have accurately stereotyped me.
aM: Okay. Now, I have to admit my parents were worried about that. And they said, “Ecstasy! Clubs!” and I was like, “No, Mom. That's a different rave. My people are not indoors. We’re outdoors, and we're not paying for stuff, and there's no bar, and there's no drinking. We're just dancing and it's daytime. It was a different thing. But that's really why I got involved in this math thing. In some sense, I wanted to know how all of that music worked, and that music was very mathematical.
EL: Oh.
aM: But then I kind of lost interest in studying the math of that because I just got involved in combinatorics and all the beautiful, theoretical math that fills my spirit and soul. But the origin story is a little bit rave, but mostly that bus.
EL: Yeah. A lot of good things happen on buses.
aM: You guys know about the art gallery theorem? Guarding a museum.
EL: Yeah. Yeah.
aM: What’s the minimum number of guards? Okay, I took the seat of someone—my postdoc was at Bowdoin college, and sadly the person who passed away shortly before I got the job was a combinatorialist named Steve Fisk (I hope I’ve got the name right). In any case, he's in the Proofs from the Book, for coming up with a proof for that art gallery theorem. You know, the famous Proofs from the Book, the idea that all the beautiful proofs are in some book? But yeah, guess where he came up with that, he told the chair of the math department when I started there: on a bus! And he was somewhere in Eastern Europe on a bus, and that's where he came up with it. And it's just like, yeah, things can happen on a bus, you know?
EL: Yeah. Now I want our listeners to, like, write in with the best math they've ever done on a bus or something. A list of bus math.
aM: You also have to include trains, I think, too.
EL: Yeah. Really long buses.
aM: All public transportation.
EL: Yeah. So something that we like to do on this podcast is ask our guests to pair their theorem with something. So what have you chosen to pair with your favorite theorem?
aM: Oh my gosh, I was supposed to think about that. Yes. Okay. Oh, 7.
EL: I feel like you have so many interests in life. You must you must have something you can think of.
aM: Oh, no, it's not a problem. I do currently a lot of mathematics. I'm in my office, sadly, a lot of hours of the day, but sometimes I leave my office and go to the pub down the road. And I call it a pub because it's really empty and brightly lit and not populated by students. It's kind of like a grown up bar. But I do a lot of recreational math there, especially on primitive roots recently. So I think I would pair my 7 theorem with seven sips of Michelob golden draft light. It's just a boring domestic beer. And then I would go across the street to the pizza place that's across from my tavern, and I would eat seven bites of a pizza with pepperoni, sausage, green pepper, and onion.
EL: Nice.
aM: I have a small appetite. So seven people would say yes, he can probably do seven bites before he’s full and needs to take a break.
EL: Or you could you could share it with seven friends.
aM: Yes. Oh, I'm often taking students down there and buying pizza for small sections of research students or groups of seven. Yes.
EL: Nice. So I know you wanted to share some other things with us on this podcast. So do you want to talk about those? Or that? I don't know exactly what form you would like to do this in.
aM: Oh, I wrote a poem. Yeah, I just want to share a poem that I wrote that maybe your listeners might find cute.
EL: Yeah. And I'd like to say I think the first time—I don't think we actually met in person that time, but the first time I saw you—was at the poetry reading at a Joint Math Meeting many years ago.
aM: Oh my gosh! I did this poem, probably.
EL: You might have. I’ll see I remember you. Many people might have seen you because you do stand out in a crowd. You know, you dress in a lot of bright colors and you have very distinctive glasses and hair and everything. So you were very memorable at the time. Yes, right now it's pink, red, and yeah, maybe just different shades of pink.
aM: Yes.
EL: But yeah, I remember seeing you do a poem at this this joint math poetry thing and then kept seeing you at various things and then we met, you know, a few years ago when I was at Eau Claire, I guess, we actually met in person then. But yeah, go ahead, please share your poem with us.
aM: Okay, this is part of the origin story again. This was just shortly after this seven thing from the bus. I was introduced to a proofs class, and they were teaching bijective functions. And I really didn't get the book. It was written by one of my teachers, and I was like, you know, I wrote a poem about it. And I think I understand my poem a little bit more than what you wrote in your book. And like, they actually sing this song now. So they recite it, so say the teachers at Sonoma State, each year to students who are taking this same course. But here it is, I think it's sometimes called a rap because I kind of dance around the room when I sing it. So it's called the Bijection Function Poem. And here you go. Are you ready?
EL: Yes.
KK: Let’s hear it.
aM: All right.
And it clearly follows that the function is bijective
Let’s take a closer look and make this more objective
It bears a certain quality – that which we call injective
A lovin’ love affair, Indeed, a one-to-one perspective.
Injection is the stuff that bonds one range to one domain
For Mr. X in the domain, only Miss Y can take his name
But if some other domain fool should try to get Miss Y’s affection,
The Horizontal Line Police are here to check for 1 to 1 Injection.
(Okay, that’s a little racy.)
Observe though, that injection does not alone grant one bijection
A function of this kind must bear Injection AND Surjection
Surjection!? What is that? Another math word gone surreal
It’s just a simple concept we call “Onto”. Here‟s the deal:
If for EVERY lady ‘y’ who walks the codomain of f
There exists at least one ‘x’ in the Domain who fancies her as his sweet best.
So hear the song that Onto sings – a simple mathful melody:
“There ain’t a Y in Codomain not imaged by some X, you see!”
So there you have it 2 conditions that define a quality.
If it’s injective and surjective, then it’s bijective, by golly!
(So this is the last verse. And there's some homework problems in my last verse, actually.)
Now if you’re paying close attention to my math-poetic verse
I reckon that you’ve noticed implications of Inverse
Inverse functions blow the same tune – They biject oh so happily
By sheer existence, inverse functions mimic Onto qualities (homework problem 1)
And per uniqueness of solution, another inverse golden rule (homework problem 2)
By gosh, that’s one-to-one & Onto straight up out the Biject School!
Word!
aM: Yeah, I never tire that one. I love teaching a proofs class.
EL: Yeah. And you said you use it in your class every time you teach it?
aM: Every time I have to say bijection. I mean, the song works, though. My only drawback in recent times is my wording long ago for “Mr. X in the domain” and “Miss Y can take his name” and the whole binary that this thing is doing. So I do have versions, I have a homosexual version, I have a this version—this is the hetero version—then I have the yet-to-be-written binary-free version, which I don't know how to make that because I was thinking for “Person X in the domain, only Person Y can take his name,” but you know person doesn't work. It's too long syllabically so I'm working on that one.
EL: Yeah.
aM: I’m working on that one.EL: Well, yeah, modernize it for for the times we live in now.
aM: Yes. I kind of dread reading and reciting this is purely hetero version, you know? And also there's not necessarily only one Miss Y that can take Mr. X’s name. I mean, you know, there's whole different relation groups these days.
EL: Yeah.
aM: But I'm talking about the injection and surjection.
EL: Yeah, the polyamorous functions are a whole different thing.
KK: Those are just relations, they’re not functions. It’s a whole thing.
aM: Oh, yes, relations aren't necessarily functions, but certain ones that be called that right?
EL: Yeah. Well, thank you so much for joining us. Is there anything else you would like to share? I mean, we often give our guests ways to find—give our listeners ways to find our guests online. So if there's anything, you know, a website, or anything you’d like to share.
aM: Can you just link my web page or should I tell you it? [Webpage link here] Actually googling “aBa UWEC math.” That's all it takes. UWEC aBa math. Whenever students can’t find our course notes, I just say like, “I don't know, Google it. There's no way you cannot find our course notes if you remember the name of your school, what you're studying and my name.” Yeah.
EL: We’ll put a link to that also in the show notes for people.
aM: Yeah, one B, aBa, for the listeners.
EL: Yes, that's right. We didn't actually—I said it was the only one spelled that way but we didn't spell it. It's aBa, and you capitalize the middle, the middle and not the first letter, right?
aM: No, yes, that's fine. It looks more symmetric that way.
EL: Yeah. You could even reverse one of them.
aM: I usually write the B backwards. Like the band, but I can't do that usually, though. I don't want to be overkill to the people that I work around. But yes, at the bottom of my webpage, I have the links to videos of me singing various songs to students, complex analysis raps, PhD level down to undergraduate level, just different raps that I wrote for funs.
And I wanted to plug one thing at JMM. I mean, not that it's hard to find it in the program, but I'm an MAA invited speaker this time, and I'm actually scared pooless a little bit to be speaking in one of those large rooms. I don't know how I got invited. But I said yes.
KK: Of course you said yes!
aM: Well, I'm excited to share two research projects that I've been doing with students. Because I like doing research just for the sheer joy of it. And I think the topic of my talk is “A research project birthed out of curiosity and joy” or something like that, because one of the projects I'm sharing wasn't even a paid research project. I just had a student that got really excited to study something I noticed in Pascal's triangle, and these tridiagonal real symmetric matrices. I mean, it was finals week, and I was like, “You want to have fun?” And we spent the next year and a half having fun, and now she's pursuing graduate school, and it's great. It's great, research for fun. But one thing I'm talking about that I'm really excited about is the Fibonacci sequence. And I know that's kind of overplayed at times, but I find it beautiful. And we're looking at the sequence modulo 10. So we're just looking at the last, the units digits.
EL: Yeah, last digits.
aM: And whenever you take the sequence mod anything, it's going to repeat. And that's an easy proof to do. And actually Lagrange knew that long, long ago. But recently, in 1960, a paper came out studying these Fibonacci sequences modulo some natural number, and proved the periodicity bit and proved—there’s tons of papers in the Fibonacci Quarterly related to this thing. But what I'm looking at in particular is a connection to astrology—which actually might clear the room, but I'm hoping not—but the sequence has a length of periods 60. So if you lay that in a circle, it repeats and every 15th value in the Fibonacci number ends in 0. That's something you can see with the sequence, but it’s a lot easier to see when you're just looking at it mod 10. and that's something probably people didn't know now. Every 15th Fibonacci number ends in 0.
KK: No, I didn't know that.
aM: And if it ends in 0, it's a 15th Fibonacci number. And so, it’s an if and only if. And every 5th Fibonacci number is a multiple of five. So in astrology, we have the cardinal signs: Aries, Cancer, Libra and Capricorn. And you and you lay those on the zeros. Those are the zeros. And then the fixed and mutable signs, like Taurus, Gemini, etc, etc. As you move after the birth of the astrological seasons, those ones lay on the fives, and then you can look at aspects between them. Actually, I'm not going to say much astrology, by the way, in this talk. So people who are listening, please still come. It's only math! But I'm going to be looking at sub-sequences, but it got inspired by some videos online that I saw by a certain astrologer. And I—there was no mathematics in the videos and I was like, “Whoa, I can fill these gaps.” And it's just beautiful. Certain sub-sequences in the Fibonacci sequence mod 10 give the Lucas sequences mod 10. The Lucas sequence, and I don't know if your listeners or you guys know what the Lucas sequence is, but it's the Fibonacci sequence, but the starting values are 2 and then 1.
KK: Right.
aM: Instead of zero and one.
EL: Yeah.
aM: And Edward Lucas is the person, actually, who named the Fibonacci sequence the Fibonacci sequence! So this is a big player. And I am really excited to introduce people to these beautiful sub-sequences that exist in this Fibonacci sequence mod 10. It's like, just so sublime, so wonderful.
EL: I guess I never thought about last digits of Fibonacci numbers before, but yeah, I hope to see that, and we'll put some information about that in the show notes too. Yeah, have a good rest of your day.
aM: All right, you too, both of you. Thank you so much for this invitation. I’m happy to be invited.
EL: Yeah, we really enjoyed it. v KK: Thanks, aBa.
aM: All right. Bye-bye.

On this episode of My Favorite Theorem, we talked with aBa Mbirika, a mathematician at the University of Wisconsin Eau Claire. He told us about several favorite theorems of the moment before zeroing in on one of his first mathematical discoveries: a way to determine whether a number is divisible by 7.

Here are some links you may find interesting after listening to the episode.

aBa’s website at UWEC
Snake lemma
Short five lemma
Euler-Fermat’s theorem
Gauss’s primitive roots
Adriana Salerno’s episode of the podcast
Steve Fisk’s “book proof” of the art gallery theorem
Information on aBa’s MAA invited address at the upcoming Joint Mathematics Meetings