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

Morning Gnomon,

Well my reasoning is that, since the

http://primes.utm.edu/largest.html

website says,


'The Sieve of Eratosthenes is still the most efficient way of finding all very small primes (e.g., those less than 1,000,000). However, most of the largest primes are found using special cases of Lagrange's Theorem from group theory. See the separate documents on proving primality for more information. '

then most of the 'many different ways of finding primes' to which you refer must be 'special cases of Lagrange's Theorem'.

But, as I say, I'm no mathematician. I was just using logic here... and it gave me an opportunity to link to the EGE which contains Langrange' Theorem.

That puts a slightly different perspective on it. What it is saying is that the really large primes (more than a thousand digits) are found using special cases of Lagrange's theorem.

There's a large region in between where there are lots of different ways, including (for six and seven digit numbers), just dividing by all the odd numbers less than the square root, which anybody can do on a spreadsheet.

Actually, I've a feeling I put <10000 in my Entry. I know that at one stage 1 had it at <1000,000 but for some reaon changed it back. I'll put it back to 1000,000.

Hey Gnomon, I'm not sure if I'm confused or not.

I've been talking about numbers less than 10,000 (or 1,000,000)

You're talking about prime numbers containing 1000 digits!!!

However, that website I've been using implies that primes below NUMBER 1,000,000 are very large.

Then in the next para it says:

In 1984 Samuel Yates defined a titanic prime to be any prime with at least 1,000 digits

....

These are the two paras in question from the website:

'The Sieve of Eratosthenes is still the most efficient way of finding all very small primes (e.g., those less than 1,000,000). However, most of the largest primes are found using special cases of Lagrange's Theorem from group theory. See the separate documents on proving primality for more information.

In 1984 Samuel Yates defined a titanic prime to be any prime with at least 1,000 digits [Yates84, Yates85]'.

I meant in Post 45 that numbers ABOVE 1,000,000 are large.

Post 65 (Go back to bed BigAl!)

I'd say the Sieve, (or a variant of it known as "Baby Divide") is useful up to about 1,000,000. (6-digit)

Really big primes are found using the Lagrange methods mentioned.

But there is a grey area in between, since they don't say what "really big" means. I know Euler proved that 2^31 - 1 was a prime using his own methods, and that's 9 digits.

I'll have to find out what the Lagrange theorem is and get back to you.

I think Fermat's theorem is used as well.

The little one, not the great: that is p is a prime number, then a^p - a is divisible by p for all integer a. (This can be proved using Lagrange's theorem, but there are other ways.)

Mostly the method is used to see that a number is not prime: if, say, 2^n - 2 is not a multiple of n, then n is not a prime.

The problem (if my memory doesn't fail me) is that there may exist composite numbers n such that a^n - a is always divisible by n. Anyway, once such an n is found, it is deemed 'good candidate for primality' and then you can try other methods to see if it is really prime or not.



You could say something like: 'for very large numbers, Eratosthenes's sieve is not very effective and more elaborate methods are required, such as the use of Lagrange's theorem from group theory'.



Also, I think that while people are looking for very large primes N, they are not looking for *all* primes up to N. This is why the new prime numbers always have a form like 2^n-1 (Mersenne numbers) or 2^{2^n}+1 (Fermat numbers).

Quite right, TB.

I've done it like that TB

How's this coming along?

Apparently, 25209047 is not a prime number.

(Would that it were!

There is still a chance that the Edited Entry will have a prime A-number though

Is everyone happy with this as it stands?

Content:

These occur at the points where y=0. Riemann also discovered something quite unexpected: the 'zeros' all seem to be in a straight line; that is, have the same value of 'z'. -- that should be, have the same value of 'x'. The z value of the zeroes is automatically zero, because that's what being a zero means.

unit of magnetic induction -- the gauss is a unit of magnetism rather than of magnetic induction. The magnetic field at any point around a stationary bar magnet can be measured in gauss, even though there is no induction happening.

Formatting and typos:

part of the set of ordinary numbers, known as integers -- I'd like to see that comma removed. As it stands, it says that the ordinary numbers are the integers, which I don't agree with. Without the comma, it is talking about "the ordinary numbers known as the integers", which implies there are other sorts of ordinary numbers.

Prof.Marcus du Sautoy -- change the full stop to a space

the next breaktrough --> the next breakthrough

fifteen year old boy --> 15-year-old boy

1777-1855 -- put spaces around the dash

(1G = 10-4 tesla) -- put a full stop after the closing bracket

newly discovered planet --> newly-discovered planet

"However, Gauss had managed to find " -- I suggest you remove the "however" from this, as there is another "however" a few sentences later. It would be better phrased as:

"Gauss was able to find"

an interest mathematics and history --> an interest in mathematics and history

sixteen years of age --> 16 years of age

They are harmonies in the music of the primes.' -- there's a quote sign at the end of that but no matching one earlier.

stray from Gaussian's distribution -- that should be "stray from the Gaussian distribution" or "stray from Gauss's distribution". I don't know which is right.

calculable probability -almost as though -- there should be a space on each side of that dash

that Fermat had achieved with his 'last theorem' -- you link to the Fermat's Last Theorem entry on the name Fermat. Wouldn't you be better to put the link on "last theorem"?

a trained, but amateur and unknown mathematician -- remove the comma or add another one after unknown

three academics at University of Cambridge --> three academics at the University of Cambridge

discovered, and therefore corroborated the work of Gauss -- add comma after corroborated

naivetéy --> naiveté

namely the Reimann Hypothesis --> namely the Riemann Hypothesis

approximately equal; to -- remove the semicolon

to characterize --> to characterise

synchronized --> synchronised

'These occur at the points where y=0. Riemann also discovered something quite unexpected: the 'zeros' all seem to be in a straight line; that is, have the same value of 'z'. -- that should be, have the same value of 'x'. The z value of the zeroes is automatically zero, because that's what being a zero means.'

I'm not sure that that's true. On a 3-D graph where x= vertical, y = horizontal and z = 'projecting out of the plane of the paper', the values of z were in a straight line.

I'll get onto your other points later. I've been laid up in bed for 3 days with some dread lurgy. Alternate hot and cold; sweating and shivering. No energy...

If by z you mean the third co-ordinate of an xyz 3-axis system, then z is the "height" and a zero is a point with z=0 (sealevel, remember?). All the points with z=0 are in a line, and that line has x=1/2 and y varying.

But to complicate things, mathematicians use z to mean a complex number, so in that notation, a z which is a zero will have its zeta function = 0 and all these z's (complex numbers) are in a line. That may be what you're thinking of.

What I said earlier is correct.