##### A Simple Explanation Of The Legendary Math Problem That Was Just Cracked

Whenever a major leap is made in pure mathematics, the question always leads to “Cool, but what does this mean for *me*?”

Here’s what you want to know.

Shinichiu Mochizuki is a number theorist at Kyoto University.

He went to Philip Exeter Academy, one of the most prestigious High Schools in the country, and graduated in a brief two years. He entered Princeton University at age sixteen and left with a Ph. D at 22. He was a full professor by 33, an absurdly young age for academia. And now, this mathematical rock star may have just cracked one of the most important problems in his field.

In short, the abc conjecture — proposed in 1985 — explores the relationships between prime numbers.

It’s been described as the most important unsolved problem in Diophantine Analysis, a branch of mathematics that — by working with some of the most simple number systems (like ax + by = 1 or x^{n} + y^{n} = z^{n}) explores some of the deepest relationships in math.

So if you’re looking for an instant “real world application,” hit the back button — but if you want to see why one equation can tell us so much about how numbers work, read on.

The abc conjecture is as follows:

*Take three positive integers that have no common factor and where a + b = c. For instance, 5, 8, and 13.*

*Now take the distinct prime factors of these integers—in this case 2, 5, and 13—and multiply them to get a new number, d.*

*In most cases, like this one, d is larger than c. The conjecture states that in rare instances where d is smaller than c, it is usually very close to c. The conjecture also shows that there are a finite number of instances where d is smaller than c. *

Mochizuki claims to have cracked this conjecture in a 500-page proof.

Even if you didn’t catch all of that, solving this would be the necessary missing link for a dozen different, more advanced problems in Diophantine Analysis, including the legendary mathematical problem *Fermat’s Last Theorem*. If Mochizuki did successfully prove it, he may have singlehandedy moved his field forward two decades.

##### Travelling salesman problem

The **travelling salesman problem** (**TSP**) is an NP-hard problem in combinatorial optimization studied in operations research and theoretical computer science. Given a list of cities and their pairwise distances, the task is to find the shortest possible route that visits each city exactly once and returns to the origin city. It is a special case of the travelling purchaser problem.

The problem was first formulated as a mathematical problem in 1930 and is one of the most intensively studied problems in optimization. It is used as a benchmark for many optimization methods. Even though the problem is computationally difficult, a large number of heuristics and exact methods are known, so that some instances with tens of thousands of cities can be solved.

The TSP has several applications even in its purest formulation, such as planning, logistics, and the manufacture of microchips. Slightly modified, it appears as a sub-problem in many areas, such as DNA sequencing. In these applications, the concept *city* represents, for example, customers, soldering points, or DNA fragments, and the concept *distance* represents travelling times or cost, or a similarity measure between DNA fragments. In many applications, additional constraints such as limited resources or time windows make the problem considerably harder.

In the theory of computational complexity, the decision version of the TSP (where, given a length *L*, the task is to decide whether any tour is shorter than *L*) belongs to the class of NP-complete problems. Thus, it is likely that the worst-case running time for any algorithm for the TSP increases exponentially with the number of cities.

The traditional lines of attack for the NP-hard problems are the following:

- Devising algorithms for finding exact solutions (they will work reasonably fast only for small problem sizes).
- Devising “suboptimal” or heuristic algorithms, i.e., algorithms that deliver either seemingly or probably good solutions, but which could not be proved to be optimal.
- Finding special cases for the problem (“subproblems”) for which either better or exact heuristics are possible.

(**Editor’s note:** I love reading this kind of stuff. Researching TSP as it applies to pointillism and printer plotting.)

##### Computing Texas Hold 'em

When he began studying poker, Rubin frequently thought in terms of how a computer might model the game. Several disciplines were applicable—game theory, expert systems, machine learning, combinatorics. The latter is a branch of mathematics concerned with finite countable structures. The various combinations of cards in a poker hand are finite countable structures. As he trained himself to be a better player, Rubin would make up combinatorics poker problems, then solve them on a computer. He has considered studying the game by creating decision trees, branching diagrams that plot a chain of if-then options and are routine for a computer scientist. For example, he could start with a single hand, then chart all the variables—his position in a round of betting, the texture of the flop (that is, does it have potential to create strong hands like straights or flushes), whether he is playing against three others or heads-up against a single remaining opponent—to see what might happen. “For any given spot in the decision tree,” he says, “I could come up with a probability distribution of different plays. Then I could write a learning program that I could use as a simulator on the computer and play a thousand times with particular settings, then tweak the settings and run it again to see if I do better, and work backward from it to infer why that was a better play in that situation. The thing is, there are so many variables and so many factors you rarely find yourself in a precise situation that you’ve studied. What you have to do is abstract out the reasoning used to get to that decision, then apply that logic and process to whatever situation you’re in.”