On Thu, Feb 14, 2019 at 11:01 AM songbird <songb...@anthive.com> wrote: > all such proofs i have ever seen are based upon the > assumptions that there are infinite numbers of such > things like primes.
I posted an abbreviated proof of that in a footnote. It's a proof by contradiction. First, assume that there are, in fact, a finite number of primes. If that's the case, then all primes must be integers between 2 and some number p, the highest prime. Take the product of all primes - call it x. When you take the product of positive integers, the result must always be at least as large as any of the factors, so x >= p. Also, x must be a multiple of every prime, which in turn means that x+1 cannot possibly be a multiple of any such prime. Thus the value x+1 must either be prime, or be the product of prime numbers that aren't in your collection of primes; therefore the collection of primes cannot possibly be complete. Therefore there are indeed an infinite number of primes. So it's not an assumption; it's a proven point. The subdividability of the universe is actually irrelevant. Perhaps the universe, at some level, becomes indivisible; but numbers don't. For any two non-equal real numbers, it is always possible to find another number in between them. (This is NOT true of floating-point numbers or any other fixed-size representation.) Numbers are actually extremely convenient like that. This is largely off-topic for Python, but do consider: thanks to bignum integers and the Fraction type, we can represent any rational number, assuming we have enough storage space. Or do we..... ChrisA -- https://mail.python.org/mailman/listinfo/python-list
