#11 --- quoting A MATHEMATICIAN'S APOLOGY, Godfrey H. Hardy, 1940,
pages 92-94 ---
    I can hardly do better than go back to the Greeks. I will state
and prove two of the famous theorems of Greek mathematics. They are
'simple' theorems, simple both in idea and in execution, but there is
no doubt at all about their being theorems of the highest class. Each
is as fresh and significant as when it was discovered-- two thousand
years have not written a wrinkle on either of them. Finally, both the
statements and the proof can be mastered in an hour by any intelligent
reader, however slender his mathematical equipment.
  1. The first is Euclid's (Elements IX 20. The real origin of many
theorems in the Elements is obscure, but there seems to be no
particular reason for supposing that this one is not Euclid's own)
proof of the existence of an infinity of prime numbers.
   The prime numbers or primes are the numbers (A)
2,3,5,7,11,13,17,19,23,29,... which cannot be resolved into smaller
factors. (There are technical reasons for not counting 1 as a prime.)
Thus 37 and 317 are prime. The primes are the material out of which all
numbers are built up by multiplication: thus 666 = 2x3x3x37. Every
number which is not prime itself is divisible by at least one prime
(usually, of course, by several). We have to prove that there are
infinitely many primes, i.e. that the series (A) never comes to an end.
   Let us suppose that it does, and that 2,3,5,..., P is the complete
series (so that P is the largest prime); and let us, on this
hypothesis, consider the number Q defined by the formula Q =
(2x3x5x..xP) + 1. It is plain that Q is not divisible by any of
2,3,5,...,P; for it leaves the remainder 1 when divided by any one of
these numbers. But, if not itself prime, it is divisible by some prime,
and therefore there is a prime (which may be Q itself) greater than any
of them. This contradicts our hypothesis, that there is no prime
greater than P; and therefore this hypothesis is false.
   The proof is by reductio ad absurdum, and reductio ad absurdum,
which Euclid loved so much, is one of a mathematician's finest weapons.
It is a far finer gambit than any chess gambit: a chess player may
offer the sacrifice of a pawn or even a piece, but a mathematician
offers the game.
--- end quoting A MATHEMATICIAN'S APOLOGY, Godfrey H. Hardy, 1940,
pages 92-94 ---

  Hardy made two mistakes. First, he felt that Euclid did a indirect
proof of IP when from the language of Euclid indicates that his method
was direct. Perhaps Hardy never tried a direct proof of IP for if he had
done so then he may have picked up on the illogic of using a prime
factor search in the indirect method when it is crucial for the direct
method and that you cannot use the prime factor search in both methods.
Secondly, when Hardy forms Q, his logic should have been up to par to
realize that Q was necessarily prime given the definition of what it
means to be prime. And thus Q discharges the initial assumption.

#12 --- quoting AN INTRODUCTION TO THE THEORY OF NUMBERS, Hardy &
Wright,
1938, p 12 ---

  The Series of Primes
2.1. First proof of Euclid's second theorem. Euclid's own proof of
Theorem 4 was as follows.
  Let 2,3,5, ..., p be the aggregrate of primes up to p, and let
  (2.1.1)  q = (2x3x5x...xp)+1.

  Then q is not divisible by any of the numbers 2,3,5,...,p. It is
therefore either prime, or divisible by a prime between p and q. In
either case there is a prime greater than p, which proves the theorem.
--- end quoting AN INTRODUCTION TO THE THEORY OF NUMBERS, Hardy &
Wright ---

Hardy and Wright fail to give a valid proof because once they formed
P!+1 it is necessarily prime and thus end of the proof. There is no
prime factor search because P!+1 has to be prime given the definition of
"primeness" at the start of the proof.

Archimedes Plutonium
www.iw.net/~a_plutonium
whole entire Universe is just one big atom where dots
of the electron-dot-cloud are galaxies