Warp wrote:
>   1) The classical proof that there are infinitely many primes is a proof by
> contradiction: Let's assume that there's a largest prime. If we multiply
> all the primes up to that largest prime and add 1, we get a number which
> is not divisible by any of the primes, and thus the assumption we made is
> false: There was a prime which is larger than the one we assumed was the
> largest.
> 
>   However, consider this: 2*3*5*7*11*13 + 1 = 30031, which is a composite
> number.
> 
>   Isn't this a contradiction to the proof? It clearly doesn't hold that the
> product of the first n primes plus 1 is a prime.
> 
>   How to explain this apparent contradiction?

The answer lies (as I look at that bit of the wikipedia article on 
primes) in that there exists a prime smaller than 30031 by which it's 
divisible that's also larger than 13, so 13 isn't the largest prime.

>   2) Assume you have an array of 24 integers. Each element of that array
> can get a value between 0 and 7. Thus the total amount of different contents
> for such as an array is 8^24, which is approximately 4*10^21.
> 
>   Now assume that you fill the array with some values and then calculate
> all the possible permutations of that array. The amount of permutations
> for 24 elements is 24!, which is approximately 6*10^23.
> 
>   Now here's the apparent paradox: The total amount of different contents is
> about 4*10^21, and naturally all those permutations should be among them as
> well. How come the total amount of permutations, 6*10^23, is way larger
> than the total amount of possible different array contents?

The total amount of permutations doesn't consider duplicate cell values. 
  It treats 77 and 77 (with the 7s being swapped) as two different numbers.

--
Tim Cook
http://empyrean.freesitespace.net

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