Prime Number Definition.

[bugzpodder]

seems that even though u post before changing signatures when u view the message it still gives you the new sig.

anywayz i want to prove this:


I've came across this reference:

The probability that a random rational number has an even denominator is 1/3 (Salamin and Gosper 1972)?

i could not find the original proof, I attempted to develop one of my own, but it lacks something in the end:

ok, first lets establish that by adding an integer c to every element of the set of all rational numbers between 0 and 1, we can establish another set of rational numbers S such that each and one of them is between two consecutive integers, namely c and c+1, with no other rational number that is not in the set S but is also bewtween c and c+1. so proving the probability over the whole set of rational numbers is the same as proving the probability over the rational numbers between 0 and 1.

Lets consider a/b, where a and b are integers. if b is odd that accounts for 1/2 of the set.

lets look at a is even and b is even and they reduce down to any/odd. any can be either even or odd.

so lets start from the any even number b. in order to give you odd/odd, numerator has to have at least as many as factor of 2 as denominator.


let f(x) denote (x/2^d)/(x/2), where d is the largest integer such that 2^d divides into x. x/2^d accounts for the number of positive integers that is less than or equal x but is divisible by 2^d. x/2 denote the number of even numbers less than or equal to x. so f(x) is the probablity of all the rational numbers between 0 and 1 that has an even denominator of x such that after reduced will have an odd denominator

btw f(x) reduces to 2/g(x)

then the probability of even/even reducing down to odd/odd is:

(f(2)+f(4)+f(6)+...+f(x))/x

as x gets really large

or

(1+1/2+1+1/4+1+...)/x

I checked this using a computer program: the above the equation should converges to 2/3 which makes the original statement true 2/3 of even/even can be reduced to even/odd -- even/even accounts for 1/4 of the set making (2/3 * 1/4) = 1/6. adding on the odd denominator 1/2 gives you 1/2+1/6 = 2/3, so a random rational number after reduced that gives u an odd denomator gives you 2/3, making a random rational number after reduced that gives u an even denominator 1/3.

can anyone provide me with a mathematical equation?

also, i am looking for another proof:

For a, b, and c any different rational numbers, then


1/(a-b)^2+1/(b-c)^2+1/(c-a)^2

is the square of a rational number (Honsberger 1991).