Prime Numbers and Random Assumptions

Yitang ZhangJordan Ellenberg over at Slate reported on Yitang Zhang and the big goings on in the field of prime number theory. There are two stories. One is that Zhang managed to prove some thousands of year old questions. The other is that he is in his 50s.

As I hope you know, a prime number is any natural number that is divisible only by itself and the number one. So 7 is prime, but 9 is not because it is divisible by 3. The first few prime numbers are: 2, 3, 5, 7, 11, 17, 19, 23, 29. Why are prime numbers interesting? Well, there are many reasons that mathematicians study them. The most fundamental reason, which you may remember from school, is that all natural numbers can be represented by a product of prime numbers. For example, 24 equals 2×2×2×3. But mostly, I think that prime numbers are interesting because they are strange and difficult.

The big news is that Zhang proved that the set of consecutive odd numbers that are prime is unbound. I know what you’re thinking, “Wow! I have no idea what you just said!” Don’t worry, it is very simple. An example of this is 5 and 7: they are consecutive odd numbers and they are prime. On the other hand, 7 and 11 are consecutive prime numbers but they are not consecutive odd numbers; 9 is in the middle.

As you can see in the list of the first 9 prime numbers above, the further you go up, the larger the gap between prime numbers. In general. But its pretty random. The question is whether as the numbers get big whether these consecutive prime numbers stop occurring. They don’t. They get less common, but they never end. Apparently, mathematicians have long assumed that to be true. That amazed me because I always assumed it was not, but then I’m not a mathematician and I’ve never been that interested in number theory.

So the result isn’t amazing. What Yitang Zhang did is prove that this was so. He also proved some other things, but I had enough difficulty getting my head around this one. And of course, Zhang’s age makes for an interesting story in a world that (wrongly it turns out) thinks that only young people are great mathematicians.

The whole subject of prime numbers has always bugged me. From my earliest days I can remember being offended by the fact that the number 2 is prime. I don’t know how common this is, but I have always personalized numbers. One small part of that is a quasi-racist belief that even numbers are just better than odd numbers. The odd numbers are contrary; they offend me. The prime numbers are even worse! So how can a nice little even number be grouped with those thugs the primes? And it’s all because 2 is a Very Small Number. In fact, the number 2 is remarkable: it is divisible by every number equal to or less than itself.

Zhang’s brilliance was to treat prime numbers as random. That may seem obvious, but it’s not. The fact is that prime numbers are not random. But apparently, treating them like random numbers provides great insights. I don’t pretend to understand it. But I feel good that not everyone making important discoveries is younger than I am.

Afterword

Here are two rather large consecutive prime numbers: 104,681 and 104,683.

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About Frank Moraes

Frank Moraes is a freelance writer and editor online and in print. He is educated as a scientist with a PhD in Atmospheric Physics. He has worked in climate science, remote sensing, throughout the computer industry, and as a college physics instructor. Find out more at About Frank Moraes.

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