--- In

primenumbers@yahoogroups.com,

Kermit Rose <kermit@...> wrote:

> what is wrong with my "Proof"

It began OK and then fizzled out.

Here is an elementary proof.

Proposition: The product prod(p>2, (p-2)/p) over primes p>2 vanishes.

Proof: It suffices to show that the contrary proposition is absurd.

Suppose that prod(p>2, (p-2)/p) did not vanish.

Then, by taking logs, we would conclude that

sum(p>2, log(p) - log(p-2)) is finite. But

log(p) - log(p-2) > 2/p. Thus

sum(p>2, 1/p) would also be finite.

Yet that is easily proven to be absurd:

http://primes.utm.edu/infinity.shtml#punchline
Hence the product prod(p>2, (p-2)/p) does indeed vanish.

Comment: By the same argument, it follows that

the product prod(p>x, (p-x)/p) over primes p > x

vanishes for all real x > 0.

David