G.H.Hardy gave an invalid proof of Infinitude of Primes in "A Mathematician's Apology"

From: Archimedes Plutonium (a_plutonium_at_iw.net)
Date: 03/29/05 

Date: Tue, 29 Mar 2005 08:17:31 -0600

--- first posted by me in the early to mid 1990s to the Internet ---

  But my business here today is to analyze Hardy's attempt to prove
Euclid's Infinitude of Primes. As in the title of this post, Hardy
failed, but the bright light is that one can see his pretty poetry. I
quote from his book.
--- 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's flaw is that Q is NECESSARILY prime and once Q is formed it is
the
end of the proof since it forms the contradiction that simultaneously
not prime and prime; or, that Q is a larger prime than P. To search for
some prime factor as Hardy does is to show that he failed and did not
understand Q, for
it ends the proof and it is the only candidate of a prime not on the
original
finite list. But Hardy made two mistakes, one he did not look at
Euclid's IP
closely for Euclid gave a direct proof of IP by increasing cardinality
of any-
and-thus-all finite sets, boosting the set of primes to infinity , and
his
second mistake was that he could not give a valid indirect proof of IP.
[lines deleted] 

---
It is worth repeating, that in the direct proof method of increasing
cardinality a prime factor search must be undertaken. But in the
indirect proof method, the moment P!+1 is formed, is the exact moment
that the contradiction arises and that P!+1 is necessarily a new prime.
So when one does the Indirect IP and starts to make a search for a
"prime factor", well the person has failed and made a invalid 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

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