## Largest Number Found!

Late last month, Professor Warren Porterdunk of Berryville University announced that he had, in fact, found the largest number.  I was lucky enough to arrange a virtual interview with him, which is summarized here.

Of course, there is no largest number.  “That is the conventional mathematical wisdom, yes,” said Porterdunk.  “But when you include trans-transcriptional numbers — that is, numbers which cannot actually be written down — well, that’s a game-changer.”  Porterdunk calls these numbers Titans, derived from the acronym TTN, which he finds apt.

So what is the largest number?  Just a single “1” followed by infinitely many zeroes.  “That’s only one representation,” said Porterdunk.  Some real numbers have more than one representation, such as $1=0.\overline9$.  The “1” followed by infinitely many zeroes — which Porterdunk calls “die Uberzahl” as an homage to Nietzsche — also has multiple presentations.  “What makes the analysis of Titans a bit tricky is that they may have uncountably infinitely many representations,” said Porterdunk.

Of course I had to ask:  What about a single “2” followed by infinitely many zeroes?  “That’s always the first question I get,” chuckled Porterdunk.  He says that the intuition which causes us to ask that question comes from the usual right-to-left algorithm for multiplying numbers.  “But the Uberzahl has no rightmost digit, so we need to throw many intuitions out the window and develop a Titan arithmetic.”

How is this done?  First, you need something called the Extendability Axiom, which postulates that (1) for a given prime number $p,$ the limit $\lim_{n\to\infty}p^n$ exists, and (2) if $p_1 then $\lim_{n\to\infty}p_1^n<\lim_{n\to\infty}p_2^n.$  Porterdunk calls numbers of the form $\lim_{n\to\infty}p^n,$ where $p$ is prime, uberprimes.

Then, enumerate the primes, and define a Titan to be a product $\prod_{n=1}^Mp_n^{k_n},$ where either $M=\infty$ or at least one of the $k_n$ is infinite.  “The trick is to find a well-ordering on the Titans,” said Porterdunk.  “Then show that the set of Titans has a unique maximal element.  Done!”

And the trick?  Consider the uberprime $P=\lim_{n\to\infty}2^n,$ and the Titan $Q=3\cdot5\cdot7\cdot11\cdots;$ that is, the product of all the odd primes. Which is larger?

You might think $Q$ is, since each odd prime is larger than 2.  On the other hand, if you look at the first two powers of 2 multiplied together, the result is larger than 3.  Then the next three together are larger than 5, the next three larger than 7, and so on.  So $P$ must be larger.  “Consequently, you must be very, very careful when well-ordering the Titans.  Pitfalls are everywhere,” remarked Porterdunk.

A staggering 263 pages of formal proof are necessary to demonstrate that the Titans do indeed have a unique maximal element.  “Yes, the devil is in the details,” commented Porterdunk.  “One sticky point is that Titans usually have a unique factorization into finite powers of primes and uberprimes.  But quantifying what “usually” means is a very delicate task.”

Another sticky point revolves around a representation theory for Titans.  Consider the Uberzahl.  At first glance, it may seem that a “1” followed by infinitely many zeros is just the product of the uberprimes $\lim_{n\to\infty}2^n$ and $\lim_{n\to\infty}5^n.$  “If it helps you to sleep at night, go ahead and think that,” said Porterdunk.  “But I can assure you, there is a bit more than that to a representation theory for Titans.”  Rather an understatement, I should say!

The next steps for Porterdunk?  “I’m working on topological induction right now,” he said.  Our discussion of this was quite involved, but I’ll try to briefly summarize.  First, create a topology on the Titans such that the set of finite products of uberprimes is dense.  Then use the well-ordering to induce a Titan metric on the space.  Next, invoke the metric to define convergence in the usual way.  Then the principle of topological induction states that if $Q=\lim_{n\to\infty}q_n$ is the limit of a sequence of Titans $q_n,$ and if some property holds for each of the $q_n,$ then the property also holds for $Q.$

“When you’ve been immersed in Titan arithmetic as I have,” said Porterdunk, “topological induction just seemed like the natural next step to take.  I think there’s a lot of potential there.”

What applications might the principle of topological induction have?  “Well, it’s very premature to make any definitive statement.  Very.  But let’s just say it might be quite interesting to several folks to know that after developing a little bit of analytic Titan theory, it may be possible to show that the Riemann hypothesis is actually equivalent to the P versus NP problem.”

Well, if that isn’t a mathematical cliffhanger, I don’t know what is!  Porterdunk said it will be a few years before he can tidy up all those details.  It definitely would be a coup for Titan arithmetic.  I’ll be sure to keep you up-to-date on this and other intriguing applications of topological induction.  Look out for the interview with Porterdunk in the New York Times next week — and don’t hesitate to visit http://www.dieuberzahl.org, where you’ll find a complete set of references and several papers to download.  I hope you’ll enjoy learning about trans-transcriptional number theory as much as I have!