The Prime Number Lottery

ecksor

A nice discussion of prime numbers and the prime number theorem.

http://plus.maths.org/issue27/features/sautoy/index.html

Capt'n Midnight

I've always liked the proof that there is no highest prime number

Multiply all known prime numbers by each other and then add one.
No matter which prime number you divide this new number by you will always have a remainder of one - ie. it is also prime [edit] or divisible by an unknown prime number, and around you go again.[/edit]

ecksor

Not quite ...

The proof is that there is an infinite number of primes, so you start with the initial assumption that there is a finite number of primes. Then do as you say and multiply by all of those primes and add one to the result. The result might be prime, or it might be composite, but either way there must be a prime divisor that isn't in the finite set of primes that we started with, and therefore we have a contradiction and our initial assumption must have been false.

Syth

For proving the infinty amount of primes, I've always imagined it as proving that if you have all the primes from 1 to n, then you can produce a new prime bigger than the biggest you had before. Just shows that you can't reach a roof.

The article was good. Hope the next one is out soon, cause it ends on a "To be continued..."

ecksor

Proof of an infinite number of even primes: http://www.geocities.com/CapeCanaveral/Lab/3550/crantz.htm

Victor

Is half the problem that we are culturally obsessed with base 10 number and can't see a pattern in base 2 numbers (nature's own )? Not that seeing patterns in base two would be easy for someone who only knew base 2, given the limits to what the eye can visualise.

http://plus.maths.org/issue27/features/sautoy/index.html

But in 1912 Littlewood, a mathematician in Cambridge, proved Gauss was wrong. However the first time that Gauss's guess underestimates the primes is for N bigger than the number of atoms in the observable universe - not a fact that experiment will ever reveal.

Do we know what point this is? Or is it one of those out there somewhere numbers?

ecksor

Originally posted by Victor
Is half the problem that we are culturally obsessed with base 10 number and can't see a pattern in base 2 numbers (nature's own )?

Why would it be any easier if we were used to base 2?

Do we know what point this is? Or is it one of those out there somewhere numbers?

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

Victor

Originally posted by ecksor
Why would it be any easier if we were used to base 2?

Perhaps, it's my prejudice against base 10 thats all - we have no natural reason to use base 10 other than it is the number of fingers and toes we have.

Originally posted by ecksor
http://mathworld.wolfram.com/SkewesNumber.html

... 8.185 x 10^370 by te Riele (1987), although Conway and Guy (1996) claim that the best current limit is 10^1167

Surely there is a slight discrepancy between these two numbers? And ultimately it is a false chase as they are merely looking for an average, not a real number?

ecksor

Originally posted by Victor
Surely there is a slight discrepancy between these two numbers? And ultimately it is a false chase as they are merely looking for an average, not a real number?

There is a large difference between the numbers, but they claim an inequality, and even being out by a number several orders of magnitude greater than the number of atoms in the universe can represent a step forward. It's like saying that the square root of 5 is less than 5 (because that's obvious), and then a while later we examine the square root function a bit better and we say that it's definitely less than 4 (which we suspected, but didn't have proof of. Suddenly we do, and that's a step forward!). After a while we really start to understand it and we say that the value is actually less than 3 and even less than 2.5

The actual number exists (and Littlewood proved this), but we don't know what it is. Maybe we'll never know. Analysis has many techniques for showing that a value lies between two other values but I personally find it a very tricky business even for simple functions.

ecksor

On a not quite related note, the most bonkers example of this sort of discrepancy between what we know and what we suspect is the example of graham's number (which was in the guinness book of records and probably still is for the largest number ever used in a proof) which is so large they needed to invent a new notation to be able to represent it. It represents an upper bound on the number of subjects from which to pick to ensure that we can always pick certain combinations (I think. That's a terrible explanation anyway.). However, the experts in the field believe that the answer is actually 6.

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

Capt'n Midnight

Originally posted by Victor
Perhaps, it's my prejudice against base 10 thats all - we have no natural reason to use base 10 other than it is the number of fingers and toes we have.
Surely there is a slight discrepancy between these two numbers? And ultimately it is a false chase as they are merely looking for an average, not a real number?

Back in dem days people didn't have have shoes, otherwise we'd be counting in octal...

The babolynians used base 60 -
nice handy divisions of 2 3 4 5 6 10 12 15 20
but it took ages to learn off your times tables...

ecksor

http://pass.maths.org.uk/issue28/features/sautoy/index.html

M@lice

I always supected that we would be better off if we had evolved with 12 fingers and toes when it comes to tables.

In national school it was always the 2,5 and 10 times tables that were the easiest, ie the divisors of 10. Wouldn't a base 12 system mean that our 2,3,4,6 and 12 times tables were easy. Now that would b handy cos its tiwce the amout of easy ones in base 10.

And base 12 couldn't be that much harder to learn than 10, If we had 12 fingers counting using base 10 would be a pain in the early days no. Do we not all learn to count with our fingers 1st.

Base 60 just sounds a bit mad. Sure its got more divisors but then why not use a base 600,6000 etc.:rolleyes:

Capt'n Midnight

Don't forget base 7 - Lbs / Stones.

Base 2 (& 8/16) is a lot easier for multiply/divide once you get used to shifting - and logs aren't too far behind.

Also it would have meant we'd have been able to count up to 102410 on our fingers

Victor

Originally posted by Capt'n Midnight
Also it would have meant we'd have been able to count up to 102410 on our fingers

Yeah, but it's a biatch to count to 32 never mind 1,048,576 (when using toes).

Victor

Morning!

I realise, although I have to admit I thought he was mixing his bases at first 102 = 210. But really count to 3210 in base 2 on your fingers!

Capt'n Midnight

D'oh - I can only count to 25610 on my fingers.

Victor

Originally posted by Capt'n Midnight
D'oh - I can only count to 25610 on my fingers.

Raid a jewellers for enough rings to keep track