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Posts Tagged ‘unsolved problems’

The aim of this post is to convince you that

   Yes you can!

Yes what? Yes you can understand mathematics. Even if you thought you couldn’t!

I am going to take you gently by the hand and lead you to the frontier of research. You will be able to understand a problem that to this day no mathematician has solved. Many good ones, even great ones, have tried to solve it.

[For more mathematics, you can also take a look at these highly recommended articles by Steve Strogatz].

Now to my story, and to the unsolved problem. As a child growing up in Reykjavik, I was fascinated by numbers. The whole idea of counting was wonderful. I could count stairs, the number of seconds I had to wait before searching for my friends in a game of hide-and-seek, the number of grains of sand on the beach … well, I didn’t count the grains, but I thought about whether and how I could do this!

That was when the question of whether there was a biggest number came up, and whether there were numbers big enough to count the grains of sand on the beach, or the stars in the sky. I later found out that Archimedes wrote a book, The Sand Reckoner, on this subject about fifty years after Euclid’s Elements. Archimedes was the Greek mathematician who jumped out of his bath shouting “Eureka!”

From this modest beginning, I always found mathematics to be fun and fascinating. How many of something or other are there? A pretty basic question, whether it has to do with stars, grains of sand, atoms, or numbers.

Let’s go back to the biggest number problem that troubled me long ago. It wasn’t too hard to solve: if you think you have the biggest number in the universe, think again. Because you can always add one to it to get a bigger number. Conclusion? There are infinitely many numbers. Plenty for the grains of sand on all the beaches, the stars in the skies, the atoms inside the stars, and plenty left over for fun, games, and mathematics!

So let’s think about something harder: the prime numbers. These are the numbers 2, 3, 5, 7, 11, 13, etc. They are the numbers that cannot be written as a number times another one. (Okay every number is one times itself, but that is not what we had in mind.) Thus 6 = 2×3 is not prime, and neither is 60 = 2x2x3x5. But the next number, 61, is prime.

Well, it is not so hard to find more prime numbers. But how many are there? Turns out that the answer is: infinitely many. The infinity was hiding in the etc! (Joke)

The existence of infinitely many prime numbers was known at least as far back as Euclid, who worked in Alexandria around 300 B.C. The proof of this fact is just a paragraph long. It is a truly elegant and beautiful proof.

Next question. You have probably noticed that there are pairs of primes like 5, 7 and 11, 13 that are spaced very closely. As closely as primes can be spaced, except for 2 and 3, which is an oddball case (no pun intended) that is never, ever repeated.

Again, it is easy to find more of these “twin primes.” For example, 17, 19. It is also easy to write a computer program to search for twin primes — just a few lines of code. A short paragraph. Elegant, just like Euclid’s proof. Wherever you look, you find twin primes, if you look hard enough — that is, run our computer program long enough.

How many twin primes are there? We don’t know! All of my mathematician friends (I have several, along with my old physics buddies from Reykjavik) think that there are infinitely many twin primes. But none of them know for certain if this is a true statement. No one has found a proof, despite centuries of looking for one.

So there you have it! You understand an unsolved problem at the frontiers of research! Time to head for the nearest cocktail party and spread the word!!

PS. There are many, many more unsolved problems. There are big problems, like the Clay Millennium Problems, one which has recently been solved by the Russian mathematician Grisha Perelman. This is the Poincaré conjecture. But my mathematician friends tell me that their field is like the wild west. Big, open frontiers. Many things to explore. Of course, that’s their thing. For now software and music are mine. But do I enjoy mathematics from the sidelines, or, to be more precise, my armchair. Time for a beer!

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