prove that
squar root x^2+y^2 is a irrational if x and y is odd , positiv integer .
thanks
Suppose that:
and that the square root of the sum of their squraes is rational, then there exist with such that:
Then:
Now consider any prime factor of and the divisibility of by this prime, and in consequence the divisibility of by the same prime.
This will show that if is rational it is an integer and so ia a square.
CB
You need both of the responses to follow what is going on. Suppose that and are integers and that is rational, then the first post shows that this is of necessity an integer and that is a square.
Then bobak shows that any square leaves remainder or when divided by , but if and are odd integers leaves remainder when divided by as does , so leaves a remainder of and so cannot be a square. but this contradicts our assumption that is rational, and si it must be irrational.
CB