As the designated faculty speaker, selected after nominations from all the various departments at the college, I made the following remarks, in full academic regalia, at the Dean’s List ceremony this evening at the College of Staten Island.

*Thank you very much to the Dean for the kind introduction.*

*I am pleased and honored to be here, amongst the elite of the **College of Staten Island, students who have made the Dean’s list **for academic accomplishment. You have proved yourselves in the **challenges of academic life and you should be proud. I am proud of **you.*

*You are our intellectual powers, engines of thought and reason. **Probably your minds are running all the time—it does seem a **little hot in here…*

*In the classic Wim Wenders film, “Wings of Desire,” some **characters are able to hear and experience the thoughts of others. And although on**e ordinarily imagines a library as a place of quiet **contemplation, in the film the library was a cacophy of **voices, a hundred trains of thought running through, each audibly **expressing the content of a person’s mind.*

*This is how I like to imagine the arena of the mind, lively and **exciting.*

*But also playful. Perhaps I’m called on to offer some advice to **you, and my advice is this: pursue an attitude of playful **curiosity about your subject, whatever it is. Play with new ideas, **explore all facets of them, going beyond whatever had been **expected of you. You will be led to vast new lands of imagination. *

*A while back, my son Horatio, in fourth grade at the time, showed me his math **homework. His teacher had said, “I am thinking of a number. It has **a 3 in the ten’s place. The digits add to 10. It is prime. What is **my number?” Can anyone solve it? Yes, 37 is a solution, and **Horatio also had found it. He asked me, “are there other **solutions?” Now perhaps an unimaginative person might think only of **two-digit numbers, and in this case, 37 is the only solution. **But with imagination, we can seek out larger possibilities. And **indeed Horatio and I worked together at the cafe and realized that **433 is another solution, as well as 631 and 1531. I checked a **table of prime numbers, and found that 10333 is also a solution, **as is 100,333. I began to wonder: are there infinitely many solutions? I had no idea **how to prove such a thing.*

*So I posted the question to one of the math Q & A sites, and it **immediately rocketed to the top, getting thousands of views. **Mathematicians all around the world were thinking about this **playful version of my son’s homework problem. Some of these m**athematicians wrote computer programs to find more and more*

*solutions; one young mathematician found all the solutions up to **one hundred million, systematically. Another found enormous **solutions, gigantic prime numbers, one with 416 digits (mostly **zeros), which were also solutions of the problem. But with regard **to whether there are infinitely many solutions or not, we still had no answer. It turns **out to be an open mathematical question; nobody knows, and the **question leads to deep number-theoretic waters. *

*Another instance of playfulness began when I was a graduate **student. A prominent visiting mathematician (Lenore Blum) gave a **talk about the theory of infinitely precise computation, **concerning computational devices able to undertake perfectly **accurate real number arithmetic. After the talk, inspired, a group **of us speculated about other infinitary notions of computability: **what could or should one do with a computational device able to **undertake infinitely many steps of computation? We played with the idea, **and over several years a theory gradually emerged. Although some of my **research colleagues had discouraged me, I ignored them and **continued to play with and develop my theory. In time, our ideas **grew into a new theory of infinitary computation. We had invented **the subject now known as infinite time Turing machines. Those **playful ideas led to a new theory that is now studied around the **world, with new masters theses written on it and Ph.D. dissertations written on it, and **conferences focused on it; our original article now has **hundreds of citations.*

*Play is not always easy. It took years of seriously hard work in the case I just mentioned; but **essential to that work was play. So please play! Explore your **ideas further, and see where they take you!*

*Thank you very much.*

I was there, and you did it beautifully. As your colleague (Associate Professor of History, Director of the American Studies Program) who has never met you, I wanted to run over to you and ask you to teach a Scientific World course on how to use math to play with your brain, with Martin Gardner books as textbooks. Just a thought. I’d like to audit that class.

Also, the number of instances must be infinite, yes? Because you could keep inserting zeros and as long as it ended 333 it would always be prime and the digits would always equal ten. Also, what about subdivisions of one? Would 10332.37 equal ten if you added up the digits, because the .3 and .7 would equal one? There are even more places this could go. (And what about negative numbers? If the number was, say, -210333, and you added up all the digits, and the negative was only on the 2 and not all the other digits, would that fit the criteria? Or would the negative HAVE to apply to all the digits?)

You are very kind. And your idea for a playful math course does indeed sound like fun, both to teach and to take!

Meanwhile, the problem about primes isn’t so easy as you suggest. Although 10333 and 100333 and even 1000333 are each prime, the pattern doesn’t continue, because 10000333 is not prime (it is 7 times 1428619), and there seem to be fewer and fewer instances of this particular pattern as you go up. You can see the online post I had referred to in my remarks at Can you answer my son’s fourth grade homework question? and there are exhaustive lists there of all solution instances up to one hundred million, and other large solutions much larger than one hundred million.

I like your idea of searching for negative or rational solutions—it seems you are taking up my advice to play! Although the usual concept of prime numbers is not usually applied to fractions (every rational number seems to be composite, since every fraction is two times its half), it would nevertheless make sense for negative numbers. But it would seem that the negative sign should apply to every digit, not just the first, and in this case we wouldn’t get 10. Hmmnnn. Let’s think about it for a bit, perhaps there is something to play with here…

Dear Joel, I read it and now I cannot stop my mind to think and my hands to write about it! The lecture is about what we lost many years ago: “the glory of childhood”, “the joy of playing like a child”, “the boundless world of a child’s imagination and questions” and “his/her endless endeavor to find the answers”. What an amazing approach to mathematics! Mathematics is not a rigid subject of science. It is a playground! So please play children! Please play….

I feel that I am 8 years old now! Carry my warm greeting to Horatio, please.

Oh! I forgot something. If you love cinema I recommend a Japanese film about “number theory”, “childhood” and “Platonism”. I hope it be enjoyable for you.

The Professor’s Beloved Equation

Director: Takashi Koizumi

Released: 2006