# Seven Prime Questions

Five of the mathematicians who meet each week for the New York Number Theory Seminar—three of whom have done so each week since they cofounded the seminar in 1982—answer (a prime number) of prime number questions below.

## What is a prime number?

Mel Nathanson, City University of New York

A prime number is a number greater than one, a whole number whose only divisors—the only numbers that divide evenly into it—are itself and one. So three is a prime but six is not because two and three both divide into six.

## Why are prime numbers fascinating?

Kevin O'Bryant, City University of New York

Because it's easy to understand what they are and hard to prove that they behave any way. And it's hard to prove that they behave the way that they should. It's hard to prove that they don't behave the way that they shouldn't. So it's like a little puzzle that you look at and you think, "Ahhh, that's easy." But the more you twist it and the more you twist it, it still looks easy, but you never get to the end. You never actually solve the problems.

## Are prime numbers important in everyday life?

David Chudnovsky, Brooklyn Polytechnic University

Prime numbers are the ultimate in simplicity. They just are divided by one and themselves. It's really like in minimalist art or minimalist sculpture. You can really produce a lot by extremely minimal means. But prime numbers are actually not only of purely academic interest. Because when you pick up your cell phone you are using prime numbers. A lot of algorithms develop in number theory for communication security and for transmission and compression of information. They depend on number theory and ultimately on prime numbers. So it is truly exploring this universe of numbers, which for a mathematician is very real [and] has application in the physical world, surprising application in the physical world.

Gregory Chudnovsky, Brooklyn Polytechnic University

Dave is right, definitely. You can find them in everyday life, they're really like elemental objects—truly elemental objects.

## What is the twin prime conjecture?

Mel Nathanson, City University of New York

Well, a twin prime conjecture is a beautiful conjecture. People have noticed for a long time that very often two consecutive odd numbers are both primes. You can't have two consecutive numbers that are primes because one will be even and one will be odd. And the even one certainly is not a prime. But you can have 17 and 19. Or 41 and 43. You just notice that the closest together two primes can be is two. Forget two and three. That's just an abnormality at the beginning. The closest together would be two. And whenever you have a pair of primes that differ by two, that's called a twin prime. And if you get a big piece of paper and start writing down the primes and looking for twin primes, or you get a big computer or the world's biggest computer and you run it looking for twin primes, you keep finding them. And everyone believes there are infinitely many pairs of twin primes, but no one's been able to prove it.

## Why is Dan Goldston's recent proof on twin primes important?

Carlos Moreno, City University of New York

Goldston's methods essentially have provided hope that we can get at the heart of the problem. It's not yet solved, but it's a major advance. It's a really significant advance. He's got a beautiful argument—it's the sort of thing that every mathematician wants to accomplish. And I think Goldston has brought back to mind the fact that no matter how hard the problems are, there is always progress in mathematics. There's continuity. There are dry times but there are also times when a lot of interesting things come up. I wouldn't be surprised if around the corner we'll see the solution to the twin prime conjecture.

## When will there be a final solution to the twin prime conjecture?

Mel Nathanson, City University of New York

You know, mathematics is very tricky. You can think you're very, very close to something, and it could take another 100 years or more or maybe just a few minutes until someone proves it. Because in mathematics you have to work hard but you also have to be clever and you have to be lucky. And sometimes you can have a very smart person with a very clever idea that you never even heard of before and, pop, there goes—he comes up with a solution. You just don't know, especially with a problem as old as this one.

## What prime numbers do you know by heart?

Mel Nathanson, City University of New York

Two, three, five, seven, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43. They keep going. 59, 61. There are a lot of them. You know, there are people who are really good at calculating in their heads. I mean, some people are phenomenal at that. They're not necessarily mathematicians, and a lot of mathematicians cannot calculate at all. We just use our fingers or do things on paper or, you know, pull out a calculator. I'm not so good at doing this stuff in my head.

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