h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

Entry: Prime Numbers - A25209047
Author: Big Al - Member, Wessex Researchers Group, Keeper of Mnemonics, Patron Saint of Left Handers - U723247

I am astounded that there are several rather 'esoteric' mathematical Entries in the EG, but none that focus on the building blocks of mathematics itself (or is it themselves? - namely Prime Numbers.So I have attempted to write it. As I am not a mathematician and can't even pretnd to understand the contributions from Riemann onwards, I have not attempted to explain the mathematics. Merely to outline the importance of the contributions to our understanding.

Wonders whether a better title might be 'Protagonists in the Search for the Prime Numbers'?

or, 'Key Steps in our Understanding of Prime Numbers'.


I just need to insert a table but I'll do that tomorrow when I'm less tired.

I'll go on reading it after breakfast.

But let it not prevent me from commenting It is a very good idea to focus on building blocks of mathematics; I also noticed exactly the same absence when writing my previous math Entries. I wanted to link to some basic mathematic entries and didn't find any

Also, being a non-mathematician is probably a good thing for clarity of the entry.

I think I might still be able to improve/clarify what I've said about the Riemann Hypothesis a tad.

Very good Entry I wish I could write them like that!



'Another way of describing a prime number is that a prime number is a positive integer that is itself not the product of two smaller positive integers.'

That's why they are also called irreducible, sometimes. Actually, this is exactly the 'irreducibility' property. The 'prime' property is the following: a number p is prime if, whenever it divides the product of 2 integers, it must divide one of the factors. By Euclid's lemma, the two notions are the same; in more general contexts you might be irreducible without being prime.

I don't know if you want to include any of this. (Maybe state the 'irreducible' terminology, which I find quite explicit.)

'(Euclid) proved (ca 300 BC) that there are...' --> There are extra parentheses around 'Euclid'.

'he kept a secret diary cum notebook' --> a secret diary-cum-notebook (with hyphens).

Gottingen --> The o should be a ö

Gauss' average curve of the 'prime number staircase' --> full stop is missing.

'if the zeros really do lie on the critical line' --> You don't say what the 'critical line' is. Nor, really, why the zeros should be located on a plane.

'one of the most famous and important unsolved problems in mathematics'

You can remind the reader that there is a 1,000,000 $ prize for the one who proves this. You could link to http://www.claymath.org/millennium/ or http://www.claymath.org/millennium/Riemann_Hypothesis/

'G H Hardy' --> GH Hardy

'to cover any mirror with a TOWEL.' --> I believe you are looking for A138232

'His reasoning was that, if God existed, He would think that Hardy hoped rain would come so that he could then get on with his work. Hardy thought that, if God existed, then He would have the sun shine all day to spite him.' --> Um, maybe some slight rephrasing here...

J E Littlewood --> JE Littlewood

'In his letter, Ramanaujan claimed...' --> Ramanujan

'E W Middlemast' --> EW Middlemast

'recognising that Ramanujan had independently and unknowingly discovered' --> I would expect a complement here. Did you mean: recognising *what* Ramanujan had discovered?

'one of his great strengths weas his mathematical naivity' --> was

' His election as a Fellow of the Royal Society was confirmed on 2 May 1918' --> full stop missing

'Turing graduated in mathematics from Cambridge in 1934,' --> the comma should be a full stop, or the sentence is incomplete.

'WWII' --> for the first occurrence, maybe you should write out 'World War II', especially since you use 'World War I' before.

'which all lay on the croitical line,' --> critical line

'Hugh Montgomery' subheader --> wouldn't you want to display his birth date too?

'they indulge in a fenetic, if brief mating orgy' --> frenetic (or frantic).



I agree that the title doesn't do justice to the Entry. There is much more than just prime numbers: with this title you might attract mostly mathematics-minded people, whereas this Entry should be of interest to a much broader public.

"Unique Factorization Theorem". Should this be 'Factorisation'?

"who lived 276 BC to 194 BC in northern Africa". 'who lived *from* 276 BC...' perhaps?

"The unit of magnetic induction, the gauss (also called 'flux density') is named after him". *The* unit of flux density (ie the SI unit) is the tesla. A gauss is 10^(-4) teslas. Although I don't suppose that really matters .

"kingdom of Hannover" --> 'Hanover'

"His father was a Lutherian minister" --> I think this should be 'Lutheran' without the 'i'

"World War 1" --> 'World War I'

There's a 'Remann' in the Turing section rather than a 'Riemann'.

Hi Big Al, haven't managed to read it all yet but will try to do so soon. Just a few things I found:



1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2, 4, 6, 6

this works fine up to the penultimate 4 (47), but the next gap would be 6, followed by 6 - I guess you accidentally repeated a 2, 4 sequence.



of Gottingen into - are you sure it is Gottingen and not Göttingen (write: Göttingen)?

Hannover is the German way to write Hanover, so it is also correct. Where did this extra 'n' go, I wonder

Don't ask. They say Munich, Nuremberg, Cologne, Hanover, so why not Gottingen?

I can understand (or maybe I'm just used to) using, say, Cologne instead of Köln - but Hanover instead of Hannover?

Did you know that the Dutch (and hence the Flemish) call 'Rijssel' the French city 'Lille'?

Excellent job, Al

I'm guessing you've read de Sautoy's 'The Music of the Primes' - an excellent book (Did you want to link to the website http://www.musicoftheprimes.com/ ?)

The only thing I think it needs is a brief description of how prime numbers underpin cryptography, and the consequences for this if anyone ever found the magic prime number formula... (I think there's an entry on cryptography you could link to.) It would make an interesting closing piece.

Could you link the word "divisible" to A23502863?

Icy

I've made the changes suggested in Posts 4, 5, and 6...

All except:

'Another way of describing a prime number is that a prime number is a positive integer that is itself not the product of two smaller positive integers.'... blah, blah, 'irreducible > Still thinking about this



'if the zeros really do lie on the critical line' --> You don't say what the 'critical line' is. Nor, really, why the zeros should be located on a plane.

As I said in Post 3, I think I can explain better what the critical line is, which I'll do. w.r.t. Why....? I don't think anyone knows; it's just that, empirically, they do.


'His reasoning was that, if God existed, He would think that Hardy hoped rain would come so that he could then get on with his work. Hardy thought that, if God existed, then He would have the sun shine all day to spite him.' --> Um, maybe some slight rephrasing here...

I had trouble phrasing this, but I'll continue to work on it


'recognising that Ramanujan had independently and unknowingly discovered' --> I would expect a complement here. Did you mean: recognising *what* Ramanujan had discovered? > I did mean what you say, but I don't quite know what you mean by 'complement'




'Hugh Montgomery' subheader --> wouldn't you want to display his birth date too? > Yes. Just havwen't yet been able to find that info.



"Unique Factorization Theorem". Should this be 'Factorisation'?

There was recent discussion in one of the threads re 'z' vs 's'. Can't rememeber what the outcome was.


"The unit of magnetic induction, the gauss (also called 'flux density') is named after him". *The* unit of flux density (ie the SI unit) is the tesla. A gauss is 10^(-4) teslas. Although I don't suppose that really matters . > I've made a minor change here.







1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2, 4, 6, 6

this works fine up to the penultimate 4 (47), but the next gap would be 6, followed by 6 - I guess you accidentally repeated a 2, 4 sequence.> Well-spotted B'El . I tried to do this late at night without writing it down first. Obviously, I have unjustified faith in my prowess

for all the comments

I can't believe I read the whole thing! Thank Bob you're *not* a mathemetician or smoke would be pouring out my ears right now.

A couple more previously unmentioned errors I picked up:

--(say, below 10, 100, 1000, 1000 etc)
>>1,000, and should the last be 10,000?

--This behaviour of the prime numbers have come to be known as
>>has come to be known (because it references the behaviour, not the numbers)

--amd invited Ramanujan to Cambridge
>>and

--Cicadas are a large group flying
>>group [of] flying

About nine years ago the 13- and 17-year cycle cicadas both turned up in the same year... What a mess it was. I didn't want to go outside for months. You couldn't see the trunks of trees for the bugs clinging to them - and forget driving around with the windows down.

I like the Protagonists title

As I said in Post 3, I think I can explain better what the critical line is, which I'll do. w.r.t. Why....? I don't think anyone knows; it's just that, empirically, they do.

No no, what I meant is: you mention the zeros without saying how they are defined (as elements z of the plane such that zeta(z)=0). They are not defined as a set of points in the plane, not very explicitly at least. This might be confusing for unsuspecting readers. Or maybe not, let's wait for other comments

I did mean what you say, but I don't quite know what you mean by 'complement'

I meant: recognising that Ramanujan had discovered a property of prime numbers, a theorem on distribution of primes, something.

Yes. Just haven't yet been able to find that info.

Ach, live mathematicians can be so vain

Milos. I've done all those

(I'm aware I've still gort a table to put in; and I I've left a '...' where I intend inserting the subject of Riemann's PhD.


Going to to have my first, though

'About nine years ago the 13- and 17-year cycle cicadas both turned up in the same year... What a mess it was. I didn't want to go outside for months. You couldn't see the trunks of trees for the bugs clinging to them - and forget driving around with the windows down.'

Going on where I stopped (I'm doing this in small steps as you can see)
>>It was a book of..... at the back of which was a table of prime numbers.<<
A book of what?

>>Riemann had observed that there were so many zeros on the critical line that it was probable that all of them are. However, he had been unable to prove it.<<

You mentioned this before and say it's called the Riemann Hypothesis, so why don't you just leave it here?


>> Early in Hardy's career he had shown that there were infinitely many zeros on the line.<<
At first, I wasn't quite sure who had shown this, maybe it would be clearer to say: 'In his career, Hardy had shown early that there were infinitely many zeros on the line.'

Btw, what line is that? I'm imagining something like a graph with an x and a y axis.

An interesting pop culture reference to Prime Numbers might be the film 'Cube'.

Btw, what line is that? I'm imagining something like a graph with an x and a y axis.

Actually, it's a graph with x,y and z axes.

You can imagine it like building a little model of a mountain on a table. It has peaks and valleys rising above and digging inside the table. The 'zeros' represent the points where the mountain model is just at the level of your table.

It looks a bit like maps of mountains, when you have 'altitude lines' - little lines following paths which are at a given altitude (do you go mountaineering?).

No, I don't go mountaineering. And don't ask me to imagine anything in 3D

I'll be getting back to this later - once I've delivered grandson to his Mum

It was a book of..... at the back of which was a table of prime numbers.<<
A book of what? >Can't remember. I need to look it up - that's why there's a '...'.


>>Riemann had observed that there were so many zeros on the critical line that it was probable that all of them are. However, he had been unable to prove it.<<

You mentioned this before and say it's called the Riemann Hypothesis, so why don't you just leave it here? > I just felt I needed to repeat it for continuity.


>> Early in Hardy's career he had shown that there were infinitely many zeros on the line.<<
At first, I wasn't quite sure who had shown this, maybe it would be clearer to say: 'In his career, Hardy had shown early that there were infinitely many zeros on the line.' Hmm, maybe Yep, see the posting that follows. This is how I'm going to achieve the clarification.

Be back in a bit

'Btw, what line is that? I'm imagining something like a graph with an x and a y axis'.

I've tried clarifying that by putting:

Riemamm also discovered something else. The 3-dimensional landscape is a 3-dimensional graph with x, y and z axes. We are mainly used to 2-dimensional graphs where the x-axis is horizontal and the y-axis is vertical. 3-dimensional graphs have an extra z-axis going backwards into the page. The Riemann landscape can be envisaged like building a 3-dimensional mountainscape model on a table. It has peaks and valleys. The Riemann'zeros' represent the points at which x=0; i.e. where the bases of the mountains are at the table level.
What Riemann also discovered was that the 'zeros' all seem to be in a straight line i.e. have the same value of 'z'. This is called the 'critical line'