Riemann zeta function (for non-math people)

[xouper]

From another thread:



1859, Berlin Academy - Bernhard Riemann published a conjecture that is today the most studied and celebrated unsolved problem in all of mathematics. A million dollar prize awaits the first person who solves it. The Riemann Hypothesis can be stated simply: "all the non-trivial zeros of the zeta function have real part one half." Understanding what that means to mathematics would fill a rather large book.

Now explain to the nice people what the zeta function is. :cool:

For the mathematically inclined:
http://en.wikipedia.org/wiki/Riemann_zeta_function
http://mathworld.wolfram.com/RiemannZetaFunction.html

A simpler introduction:
http://www.maths.ex.ac.uk/~mwatkins/zeta/devlin.pdf
It occurred to me that perhaps it might be possible to explain the zeta function to a non-math audience, at least to the point where they can get a taste of what it's about. I would be willing to give it a try if anyone is interested.

[Skep]

I have read Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics (http://www.amazon.com/exec/obidos/tg/detail/-/0309085497/103-4727041-3727061?v=glance) by John Derbyshire twice and have a better understanding of the zeta function from reading the book than from anything I was able to find on the web. In particular, Derbyshire makes the connection between the zeta function and the distribution of the primes (which is very subtle and deep) very clear. It is helpful to have a rudimentary understanding of some concepts in calculus but where it is needed, Derbyshire introduces the concepts and makes sure the reader understands before proceeding. The biographical chapters on Riemann are also very good. Excellent book; I highly recommend it.

xouper, have you read this book? Any comments? And by all means, explain away. :popcorn:

[Blake]

I was hoping you'd offer after that tantalizing mention! I wanna know.

[livius drusus]

Sign me up, too.

[Dragar]

Ditto. Though I'm not really a non-Maths person, I know next to nothing about this function. So I'd like to hear.

[wei yau]

Learning is one of the reasons why I'm here. I'll be keeping an eye on this thread.

[viscousmemories]

Whatdya want, a bribe? :)

[wade-w]

Looks like you got your answer, xouper! I'm looking forward to the discussion. Good luck; I don't think I could do it, but I'll try and help you as much as I can.

P.S.: I am a "math person."

[xouper]

OK, I'll start writing something. Stay tuned. :)

[seebs]

I'm a pseudo-math person. The other one I always wanted to learn was the Fast Fourier Tranform. My father always wanted to coach me through inventing it, but tragically, complications from diabetes killed him off before we got around to it. (He used to like, not so much teaching me mathematical things, as encouraging me to invent them independently. It turns out most math is "easy" to invent if you have a really, really, good teacher asking the right questions.)

[Dragar]

I'm studying Fourier Transforms now, seebs. Is there difference between 'fast' transforms and basic transforms?

[Ensign Steve]

It turns out most math is "easy" to invent if you have a really, really, good teacher asking the right questions.)

True dat. That was the only way I could make sense of calculus.

[seebs]

I'm studying Fourier Transforms now, seebs. Is there difference between 'fast' transforms and basic transforms?

Dunno.

[slimshady2357]

If anyone is interested in getting started before xouper gets back, I would highly recommend The music of the Primes by Marcus du Sautoy.
It's an excellent read and quite accessible.

It contains a general history of the primes, but focuses on Riemann's hypothesis and what it would mean for it to be proved.

A great read.

Adam

[Goliath]

Right now, I'm (slowly) reading my way through The Riemann Hypothesis: The Greatest Unsolved Problem in Mathematics (http://www.amazon.com/exec/obidos/tg/detail/-/0374250073/ref=pd_sbs_b_1/102-6119740-0724122?%5Fencoding=UTF8&v=glance) by Karl Sabbagh (I'm about 2/3 of the way through it). It's a very good read for the mathematician and the layperson alike, as there isn't too much mathematics as to scare the layperson off, and there is enough historical detail and conversations with people currently working on the Riemann Hypothesis to keep the math geeks interested. I haven't read Prime Obsession yet, but it's on my "to read" list.


Is there difference between 'fast' transforms and basic transforms?


Well, are we talking about :autobot: or :deceptic: ? :D

Okay, seriously, neither Fourier transforms nor the Riemann Hypothesis are exactly my cup of tea, and neither are things that I know terribly much about. I can try to scrap together what I know about the Riemann Hypothesis, though, if you folks would like.

[Dragar]

Before I forget:

Fourier transforms are, as I have been taught them, are a way of writing functions which you would normally not be able to write (such as the 'top hat' function - a box sat on the x-axis) by using infinite sums of sine or cosine waves.

Is this ringing any bells, seebs?

[ceptimus]

Fourier transforms swap from the time domain to the frequency domain and vice versa.

They're implemented real time in music playing devices nowadays, so that we can see those nifty 'frequency spectrum' displays of whatever music is playing.

Fast Fourier transforms (FFT) are a neat way of taking discrete chunks (of sound say) and doing quick transforms on them. Because a real chunk will usually not start and finish at the same zero level, you have to tweak the amplitude envelope to reduce the errors the the FFT produces. When I coded some real time FFTs years ago, I multiplied the raw values by a Hanning or Hamming function (I forget which) to shape the envelope prior to performing the FFT.

[seebs]

Yup, that's it. My dad wanted to coach me through reinventing the Fourier Transform; apparently he thought it would be a fun way to spend an afternoon.

[wildernesse]

Is the explanation of the zeta function going to include what "real world" use, if any, it has? I take a broad meaning of "real world" use.

Side note:
RA is working with the zeta function right now--some mumbo jumbo about power law distribution for his application on gap size distribution in sequence data. He's annoyed with me because I've just asked about two dozen "why?" questions regarding this thread so now he won't tell me what the sentence I typed means. But I accept that as a real world use once someone tells me what that means.

[xouper]

Is the explanation of the zeta function going to include what "real world" use, if any, it has?
I wasn't planning to go beyond explaining what the zeta function is, in very simple terms.

(pun intended :P)

Those who have read any of the three books mentioned in this thread already know more than is addressed by the explanation I'm writing. My humble little piece is more like "Zeta Functions for Dummies"®, intended for the kind of people whose eyes glaze over if they tried to read any of the three books cited above.

Do I need to rethink who my target audience is here?

[viscousmemories]

I think you're on target, xoup. You can rest assured my eyes glaze over just thinking about this, and I'm pretty sure I'm speaking for at least a couple of others in this thread. :)

[wildernesse]

Is the explanation of the zeta function going to include what "real world" use, if any, it has?
I wasn't planning to go beyond explaining what the zeta function is, in very simple terms.

(pun intended :P)

No problem. I can ask two hundred more "Why?" questions from RA and he'll probably explain at least what that sentence I typed means (and maybe the pun). :cool:

ETA: On second thought, I wanted to further explain what I meant by "real world" use--like a sentence that says "People use the zeta function when they are calculating junk in genetics" not what the junk is, necessarily. Real world use, as opposed to "because it's there". My understanding of math is based on "because it's there"--I had to learn math past basic algebra because it was there and not because math was useful. I'm only now getting over the idea that math past 8th grade algebra is more than some personal challenge, akin to deciding to take up the banjo.

Those who have read any of the three books mentioned in this thread already know more than is addressed by the explanation I'm writing. My humble little piece is more like "Zeta Functions for Dummies"®, intended for the kind of people whose eyes glaze over if they tried to read any of the three books cited above.

Do I need to rethink who my target audience is here?

Um, I may be below your target audience in understanding, so don't change anything on my account.

[livius drusus]

Bumpage, just in case xoup thinks we're letting him off the hook. ;)

[ceptimus]

I'll have a stab at it, using only English and a few numbers. Who was it who said that each equation you use loses you half of your remaining readers?

This is very informal though (and even plain wrong in some respects). I'm sure xouper's treatment will be much more illuminating...

The Reimann Zeta function is very interesting as it relates one series of numbers, the integers: 1, 2, 3, 4, 5, ... to another series of numbers, the primes: 2, 3, 5, 7, 11, ...

A prime number is one that can't be divided into equal whole numbers. 111 isn't a prime number, for example, because it is 37 multiplied by 3. Prime numbers are an important part of mathematics. Although it is dangerous to justify the study of mathematics for mere practical uses, one practical use of prime numbers is the encryption that allows you to make secure financial transactions over the Internet.

Prime numbers have many fascinating properties (to mathematicians) and there are many unsolved problems concerning them. For example, one famous theory is that every even number (other than 2) can be written as two prime numbers added together. So, for example, 12 is 5 plus 7, and 30 is 17 plus 13. No one has ever found an even number that can't be made by adding two primes, but no one has proved that such an even number doesn't exist (it would have to be very large though, as computers have checked all the even numbers of 'reasonable' size). There is a large cash prize available to anyone who can solve this conundrum.

Prime numbers don't follow any obvious pattern, and there is no known formula that is guaranteed to generate a prime number on demand. This is why the Reimann Zeta function is so tantalising. It links the integers (normal counting numbers) to the primes, and so hints at a way that prime numbers might be generated to order...

There is a snag though. The Riemann Zeta function is infinite: one side of the equation is infinity, calculated by adding simple functions of all the counting numbers (integers) together; the other side of the equation is also infinite, this time calculated by multiplying simple functions of all the prime numbers together.

A function that says 'infinity equals infinity' doesn't look that promising at first, but mathematicians are tenacious beasts, and they are able to extract useful data from this seemingly intractable equation.

And I'll leave it at that... This was merely a warm up act, with the main feature still to come... :popcorn:

[Ensign Steve]

YAY! Well done! Looking forward to the rest. :popcorn:

[Dragar]

Really interesting. Looking forward to the full show!

[xouper]

ceptimus: I'll have a stab at it, ...
I guess this means that if I don't hurry up and finish my little article, there won't be anything left to tell. Maybe I should trim it back some just for the sake of getting something posted.


ceptimus: Prime numbers don't follow any obvious pattern, and there is no known formula that is guaranteed to generate a prime number on demand. This is why the Reimann Zeta function is so tantalising. It links the integers (normal counting numbers) to the primes, and so hints at a way that prime numbers might be generated to order...
I commented on this in another thread.



wildernesse: On second thought, I wanted to further explain what I meant by "real world" use--like a sentence that says "People use the zeta function when they are calculating junk in genetics" not what the junk is, necessarily. Real world use, as opposed to "because it's there". My understanding of math is based on "because it's there"--I had to learn math past basic algebra because it was there and not because math was useful.
In my article, I had written a little sidebar that included what I thought was a clever use of "because it's there", but now that you've said this, it kinda pulls the rug out from my use of it. I only mention this because when I finally get around to posting my piece, it will look like I hadn't taken your comment to heart, even though I wrote it before you posted. Oh well, so it goes.

[Ensign Steve]

wildernesse: On second thought, I wanted to further explain what I meant by "real world" use--like a sentence that says "People use the zeta function when they are calculating junk in genetics" not what the junk is, necessarily. Real world use, as opposed to "because it's there". My understanding of math is based on "because it's there"--I had to learn math past basic algebra because it was there and not because math was useful.

WHA?! How is it used in genetics? As much as I love math for math's sake, I've been on a "so what?" kick recently and this is the first I've heard of a genetics application. Do tell!

[xouper]

Here's a teaser, something I posted on another forum not so long ago, a relationship between Pi and all the prime numbers, courtesy of the zeta function:


http://www.xoup.net/img/piprimes.gif

[Ensign Steve]

That's pretty, but am I thick? I read that and all I can think is:

What is special about the relationship between pi and 6 (or between pi-squared-minus-one and six) that makes that true?

[wildernesse]

Will anyone be offended if I talk through this--and I'll do the same thing for xoupers too, because by then I will have turned this over in my head once or twice? I cannot visualize what numbers or formulas are trying to say--it's like looking at a word for a long time or saying it a bunch of times and then it sounding weird--for me, math explanations start out with the sounding weird part or feeling, I have to really work at it to make sense. (The most awful thing is that some judges write this way--really long sentences and strange word choice that somehow make sense, but you have to work at that one sentence at a time to wrestle the meaning out of it.)


The Reimann Zeta function is very interesting as it relates one series of numbers, the integers: 1, 2, 3, 4, 5, ... to another series of numbers, the primes: 2, 3, 5, 7, 11, ...

So, it is about a relationship. (Yes, I know you said this explanation was very simplified.)

A prime number is one that can't be divided into equal whole numbers. 111 isn't a prime number, for example, because it is 37 multiplied by 3. Prime numbers are an important part of mathematics. Although it is dangerous to justify the study of mathematics for mere practical uses, one practical use of prime numbers is the encryption that allows you to make secure financial transactions over the Internet.

I thought a 2 is a prime number because its only factors (is that the right word) are 2 and 1. I'm not getting the equal part--2 and 1 seem to be about as equal as 37 and 3.

As for the dangerous part about wanting "mere" practical uses--I do understand that a good part of anything academic will and should be the pursuit of knowledge for its own sake. That's fine. However--in my experience, very, very little of my math education was focused on how or why we would use any of it. Math beyond 8th grade was a mere personal interest/challenge--not any different from having knitting required. Yeah, some people like knitting and are talented at it. It's useful to make sweaters, but if you don't ever want to make sweaters much less frilly little nothings that clutter up your house then it would be nothing more than a hoop to jump through.

Ok, I was going to make a comment about geometry and how it felt at the time incredibly useless in this same way--and remains so today. However, I realized my example was something I do use and understand so I have to revise my comment. I've moved up a grade, and now past 9th grade, math is nothing more than a personal challenge.

What prompted me to ask for something reality-based is that in my applied stat class last semester, which I really enjoyed, our professor would mention as he explained things that such and such was beyond our class, and we would be given certain numbers because we wouldn't be learning to calculate them. What wowed me is that I realized that I had been taught how to do those calculations--but had never been taught why on earth anyone would do them except to pass a math test. Ever since then, I've been really annoyed that I learned math in a way that focused on the how and not the why.

Prime numbers have many fascinating properties (to mathematicians) and there are many unsolved problems concerning them. For example, one famous theory is that every even number (other than 2) can be written as two prime numbers added together. So, for example, 12 is 5 plus 7, and 30 is 17 plus 13. No one has ever found an even number that can't be made by adding two primes, but no one has proved that such an even number doesn't exist (it would have to be very large though, as computers have checked all the even numbers of 'reasonable' size). There is a large cash prize available to anyone who can solve this conundrum.

This counts as a neat fact. No one has proved that such an even number doesn't exist--so it could exist, somewhere out in the ether. (Math exists in outer space with stars, so obviously that's where this number lives. IMO, of course.)

Prime numbers don't follow any obvious pattern, and there is no known formula that is guaranteed to generate a prime number on demand. This is why the Reimann Zeta function is so tantalising. It links the integers (normal counting numbers) to the primes, and so hints at a way that prime numbers might be generated to order...

Ok. The function relates integers and primes in a way that hints at how primes might be "made to order"--how does a function hint? What do you mean by that?

There is a snag though. The Riemann Zeta function is infinite: one side of the equation is infinity, calculated by adding simple functions of all the counting numbers (integers) together; the other side of the equation is also infinite, this time calculated by multiplying simple functions of all the prime numbers together.

Adds integers, multiplies primes. To infinity. Ok.

A function that says 'infinity equals infinity' doesn't look that promising at first, but mathematicians are tenacious beasts, and they are able to extract useful data from this seemingly intractable equation.

And I'll leave it at that... This was merely a warm up act, with the main feature still to come... :popcorn:

So they get useful data from this--and use it to....(try to find "made to order" primes?)

Thanks for the warm up, I look forward to the rest of everyone's explanations. And I'll try to either explain what RA's doing, or make him come and explain it. IF this is really annoying, I can always move to another thread.

[seebs]

I think the point is, if you have a function that, given an integer N, can give you the Nth prime... You no longer have to search for primes, just put a very large number into your function.

[viscousmemories]

I thought a 2 is a prime number because its only factors (is that the right word) are 2 and 1. I'm not getting the equal part--2 and 1 seem to be about as equal as 37 and 3.
I'm learning this as we go too. :)

You're right, 2 is prime because its only factors are 2 and 1. Any number larger than 2 that can be divided into equal whole numbers is not prime. In ceptimus' example he uses 111 as a number that isn't prime. This is because it can be divided into 3 equal whole numbers: 37 + 37 + 37 (or as he said, 37x3).

I think. :)

[ceptimus]

That's right. If a number can only be divided equally by itself and 1 then it is a prime number. So, for example, 17 is prime because if you wish to write it as a multiplication of whole numbers, there is essentially only one way to do it:

17 = 1 x 17 = 17 x 1 = -1 x -17 = -17 x -1

With a non prime number (which is called a composite number), like 15, you can use all the arrangements of 1, 15, -1 and -15, as above, but also make the number by multiplying 3 and 5.

2 is the only even prime number. Obviously, all other even numbers will divide by 2.

Zero and one are interesting cases. Are they prime or not? It is accepted, almost by definition, that zero and one are not prime, although they are clearly not composite either.

[Ensign Steve]

'Twas explained to me that 1 is not prime because it has only 1 factor. Prime numbers have exactly 2 factors. At least, that is a definition of prime that made it easy to explain away 1. ;)

Doesn't 0 have an infitite number of factors? 0/1=0, 0/2=0 ... Or does dividing into something 0 times not count as dividing evenly?

[JoeP]

I'll have a stab at it, using only English and a few numbers. Who was it who said that each equation you use loses you half of your remaining readers?
"Someone." See Stephen Hawking's acknowledgements in A Brief History of Time.

[wildernesse]

I think the point is, if you have a function that, given an integer N, can give you the Nth prime... You no longer have to search for primes, just put a very large number into your function.

So the end result of the function is a prime number, related to the integers you used? And the hinting is how the function does that?

vm, your explanation of the equal part helps.

In my article, I had written a little sidebar that included what I thought was a clever use of "because it's there", but now that you've said this, it kinda pulls the rug out from my use of it. I only mention this because when I finally get around to posting my piece, it will look like I hadn't taken your comment to heart, even though I wrote it before you posted. Oh well, so it goes.

I wouldn't worry too much about it--I realize that to some people parts of math are beautiful, elegant, etc. and that part of doing it is because it's there. That's great--but if there is a somewhat practical use, then that's what will interest me. "Because it's there" is a perfectly good reason for a person to decide to do something, but it's not necessarily the most convincing argument to interest people who don't share your interest.

[xouper]

wildernesse: ... "Because it's there" is a perfectly good reason for a person to decide to do something, but it's not necessarily the most convincing argument to interest people who don't share your interest.
Agreed. I may as well post the sidebar I wrote in one of the first drafts of my piece:

One might legitimately ask why anyone would want to do that in the first place, and the answer is the same as given by George Mallory, "Because it's there."

To a mathematician, it's a puzzle that begs for a solution.

Another good example is the square root of negative one. Of what practical use is that? None, that anyone could see at first. Mathematicians liked it for its own sake (its mathematical properties) and used it to develop numerous mathematical tools. Later someone noticed that some of those tools could be used to simplify certain aspects of electrical engineering, for example. Often a practical application becomes evident only after the mathematician has done his work.

We will be discussing the square root of negative one as part of the Riemann Zeta Function, so this topic will come up again.

I certainly have no interest in climbing Mt. Everest, but I have been known to jump out of perfectly good airplanes. :)

[Goliath]

I've been keeping an eye on this discussion with some interest...I'm a Commutative Algebraist by trade, and not a Number Theorist, so I don't know terribly much about the Riemann Zeta function, myself (except that, if the Riemann Hypothesis is true, it can tell you how the primes are distributed amongst the integers).

However, I have to take exception to this:

Zero and one are interesting cases. Are they prime or not? It is accepted, almost by definition, that zero and one are not prime, although they are clearly not composite either.

Actually, in my nook of mathematics (and in what Number Theory I've seen), zero is considered to be a prime integer, but 1 isn't. If you'd like, I could give the Ring Theoretic explanation as to why.

[ceptimus]

In my nook of mathematics (and in what Number Theory I've seen), zero is considered to be a prime integer, but 1 isn't. If you'd like, I could give the Ring Theoretic explanation as to why.
Yes please. Try and make the explanation accessible to non-mathematicians, if possible, though. :)

[Goliath]

Since there still seems to be a bit of confusion about the Riemann Zeta function (a fair amount of it being my own, let me assure you), let me try an explanation that would make most Number Theorists point and laugh at me:

A fact that you were told when you were a child, learning your multiplication tables, was that every integer except 0, 1, or -1 factors uniquely* into primes.** So, the primes are, in a sense, the "building blocks" of the integers.

About 2,500 years ago, Euclid proved in The Elements that there are infinitely many primes. So a question that is both a cool mathematical puzzle for its own sake, as well as one that has "real world" applications (such as encryption) is the following: Given a prime integer, when do you get to the next one? In 1859, the German mathematician Bernhard Riemann, wrote a paper in which he hypothesized that a certain function (which became known as the Riemann Zeta function) would be able to tell us exactly what the nth prime is for any given integer n (where the first prime is 2, the second primes is 3, and so on). So, for example, if you wanted to know what the 12,539,869,879,823,959,823,798,579,692,837,958,697th prime is, you could just input that gawd-awful number into this function, and it would give you the prime that is the 12,539,869,879,823,959,823,798,579,692,837,958,697th down the line.

However, we still don't know if the Riemann Hypothesis is correct (that is, if the function performs as promised). The Riemann Hypothesis is arguably the most infamous open problem in mathematics today.

Well, now that I've bored the living snot out of you, I'll go back to looking at the contents of my slow-cooker and wishing that it was 7:00. :hungry:

* - By "uniquely" I mean up to order and up to multiplying by some -1's. For example, 18=2*3*3, but if I were a smart-ass, I could say "Yeah, but 18=(-3)*(-2)*3, too! What about that?! Huh!? Huh?!" But I would, indeed be a smart-ass, since that's really the same factorization as 2*3*3.

** - Now, there are places other than the integers where factorization isn't so rosy...where not everything factors uniquely, or where there are things that don't factor into primes at all! That's Factorization Theory, however, and not Number Theory.

[Dragar]

Thanks Goliath. Even moderately drunk (a rarity, let me assure you), at 1am, that was really interesting. :)

[xouper]

Goliath: ... Given a prime integer, when do you get to the next one? In 1859, the German mathematician Bernhard Riemann, wrote a paper in which he hypothesized that a certain function (which became known as the Riemann Zeta function) would be able to tell us exactly what the nth prime is for any given integer n (where the first prime is 2, the second primes is 3, and so on).
I asked about this in another thread but I'll ask again here. How does the Riemann Zeta Function (or the Riemann Hypothesis) give the nth prime?

[xouper]

ceptimus: Zero and one are interesting cases. Are they prime or not? It is accepted, almost by definition, that zero and one are not prime, although they are clearly not composite either.
The second paragraph at this link has an interesting commentary on that.

http://mathworld.wolfram.com/PrimeNumber.html


The number 1 is a special case which is considered neither prime nor composite (Wells 1986, p. 31). Although the number 1 used to be considered a prime (Goldbach 1742; Lehmer 1909; Lehmer 1914; Hardy and Wright 1979, p. 11; Gardner 1984, pp. 86-87; Sloane and Plouffe 1995, p. 33; Hardy 1999, p. 46), it requires special treatment in so many definitions and applications involving primes greater than or equal to 2 that it is usually placed into a class of its own. A good reason not to call 1 a prime number is that if 1 were prime, then the statement of the fundamental theorem of arithmetic would have to be modified since "in exactly one way" would be false because any n=n*1. In other words, unique factorization into a product of primes would fail if the primes included 1. A slightly less illuminating but mathematically correct reason is noted by Tietze (1965, p. 2), who states "Why is the number 1 made an exception? This is a problem that schoolboys often argue about, but since it is a question of definition, it is not arguable." As more simply noted by Derbyshire (2004, p. 33), "On balance, 2 pays its way [as a prime] on balance; 1 doesn't."

[ceptimus]

As numbers become larger, there is a greater chance of them being composite, so the frequency at which primes occur becomes less and less. One might think that there would be a 'greatest prime' and that all numbers larger than that would be composite, but no, a famous and elegant proof by Euclid shows that there are an infinity of primes.

Say we wish to find a run of consecutive integers that don't include a prime. How might we do that? It turns out that this is quite easy. Say we want 6 consecutive non-prime numbers, first we calculate:

7 x 6 x 5 x 4 x 3 x 2 x 1 = 5040

The process of multiplying all the integers up to a certain number together, is used so frequently in mathematics (especially in statistics) that it has a special name, factorial, and the calculation above can be shortened to 'seven factorial equals 5040'. The exclamation mark is used to denote this operation, so we can write:

7! = 5040

Many calculators have a ! button, and will happily calculate numbers up to 69! for you, though most of them struggle beyond 69!

Anyway, as 5040 was made by multiplying 2 by some other numbers it is clearly divisible by 2 and so if we add another 2 making 5042, that also must divide by 2.

Similarly 5040 is divisible by 3, and so 5043 also divides by 3.

Applying the same argument repeatedly we can see that:

5042 divides by 2
5043 divides by 3
5044 divides by 4
5045 divides by 5
5046 divides by 6
5047 divides by 7

...and so we have found our run of 6 consecutive integers that are composite: they start at 7! + 2. Now it so happens that 5041 and also 5048, 5049, and 5050 are composite, so there are actually eleven consecutive composites, beginning with 5050, but the argument doesn't prove that - it only proves that the six numbers beginning at 7! + 2 are composite.

Now we can easily extend that argument - say we want ten billion consecutive integers that don't include a prime - that's easy - the ten billion integers beginning at:

10,000,000,001! + 2

are guaranteed to be composite.

Of course, the ten billion numbers beginning with 10,000,000,001! + 2 are quite formidable numbers, well beyond the capacity of current day computers to test for primality - but we don't need to test them, as we have proved they are not prime by means of a simple argument.