Wolfram's MathWorld fails to give a valid proof of Euclid's Infinitude of Primes

From: Archimedes Plutonium (a_plutonium_at_iw.net)
Date: 03/27/05 

Date: Sun, 27 Mar 2005 12:52:11 -0600

--- quoting the website page
http://mathworld.wolfram.com/EuclidsTheorems.html ---
Euclid's Theorems

                      A theorem sometimes called "Euclid's first
theorem" or
                      Euclid's principle states that if p is a prime and
,
                      then or (where means divides). A corollary is that

                      (Conway and Guy 1996). The fundamental theorem of
                      arithmetic is another corollary (Hardy and Wright
                      1979).

                      Euclid's second theorem states that the number of
                      primes is infinite. This theorem, also called the
                      infinitude of primes theorem, was proved by Euclid

                      Eric Weisstein's World of Biography in Proposition

                      IX.20 of the Elements (Tietze 1965, pp. 7-9).
                      Ribenboim (1989) gives nine (and a half) proofs of

                      this theorem. Euclid's elegant proof proceeds as
                      follows. Given a finite sequence of consecutive
primes
                      2, 3, 5, ..., p, the number
                      (1)

                      known as the ith Euclid number when is the ith
prime,
                      is either a new prime or the product of primes. If
N
                      is a prime, then it must be greater than the
previous
                      primes, since one plus the product of primes must
be
                      greater than each prime composing the product.
Now, if
                      N is a product of primes, then at least one of the

                      primes must be greater than p. This can be shown
as
                      follows.

                      If N is composite and has no prime factors greater

                      than p, then one of its factors (say F) must be
one of
                      the primes in the sequence, 2, 3, 5, ..., p. It
                      therefore divides the product . However, since it
is a
                      factor of N, it also divides N. But a number which

                      divides two numbers a and b < a also divides their

                      difference , so F must also divide
                      (2)

                      However, in order to divide 1, F must be 1, which
is
                      contrary to the assumption that it is a prime in
the
                      sequence 2, 3, 5, .... It therefore follows that
if N
                      is composite, it has at least one factor greater
than
                      p. Since N is either a prime greater than p or
                      contains a prime factor greater than p, a prime
larger
                      than the largest in the finite sequence can always
be
                      found, so there are an infinite number of primes.
                      Hardy Eric Weisstein's World of Biography (1967)
                      remarks that this proof is "as fresh and
significant
                      as when it was discovered" so that "two thousand
years
                      have not written a wrinkle" on it.

                      A similar argument shows that and
                      (3)

                      must be either prime or be divisible by a prime .
                      Kummer Eric Weisstein's World of Biography used a
                      variation of this proof, which is also a proof by
                      contradiction. It assumes that there exist only a
                      finite number of primes , , ..., . Now define and
                      consider . It must be a product of primes, so it
has a
                      prime divisor in common with N. Therefore, which
is
                      nonsense, so we have proved the initial assumption
is
                      wrong by contradiction.

                      It is also true that there are runs of composite
                      numbers which are arbitrarily long. This can be
seen
                      by defining
                      (4)

                      where is a factorial. Then the consecutive numbers
, ,
                      ..., are composite, since
                      (5)
                      (6)
                      (7)

                      Guy (1981, 1988) points out that while is not
                      necessarily prime, letting q be the next prime
after ,
                      the number is almost always a prime, although it
has
                      not been proven that this must always be the case.

                      Divide, Euclid Number, Prime Number

                      Links search

                      References

                      Ball, W. W. R. and Coxeter, H. S. M. Mathematical
                      Recreations and Essays, 13th ed. New York: Dover,
p.
                      60, 1987.

                      Conway, J. H. and Guy, R. K. "There are Always New

                      Primes!" In The Book of Numbers. New York:
                      Springer-Verlag, pp. 133-134, 1996.

                      Cosgrave, J. B. "A Remark on Euclid's Proof of the

                      Infinitude of Primes." Amer. Math. Monthly 96,
                      339-341, 1989.

                      Courant, R. and Robbins, H. What Is Mathematics?:
An
                      Elementary Approach to Ideas and Methods, 2nd ed.
                      Oxford, England: Oxford University Press, p. 22,
1996.

                      Derbyshire, J. Prime Obsession: Bernhard Riemann
and
                      the Greatest Unsolved Problem in Mathematics. New
                      York: Penguin, p. 34, 2004.

                      Dunham, W. "Great Theorem: The Infinitude of
Primes."
                      Journey through Genius: The Great Theorems of
                      Mathematics. New York: Wiley, pp. 73-75, 1990.

                      Flannery, S. and Flannery, D. In Code: A
Mathematical
                      Journey. London: Profile Books, pp. 42-43, 2000.

                      Guy, R. K. §A12 in Unsolved Problems in Number
Theory.
                      New York: Springer-Verlag, 1981.

                      Guy, R. K. "The Strong Law of Small Numbers."
Amer.
                      Math. Monthly 95, 697-712, 1988.

                      Hardy, G. H. A Mathematician's Apology. Cambridge,

                      England: Cambridge University Press, 1992.

                      Havil, J. Gamma: Exploring Euler's Constant.
                      Princeton, NJ: Princeton University Press, p. 28,
                      2003.

                      Ribenboim, P. The New Book of Prime Number
Records.
                      New York: Springer-Verlag, pp. 3-12, 1989.

                      Tietze, H. Famous Problems of Mathematics: Solved
and
                      Unsolved Mathematics Problems from Antiquity to
Modern
                      Times. New York: Graylock Press, pp. 7-9, 1965.

                      cite this as
                      Eric W. Weisstein. "Euclid's Theorems." From
                      MathWorld--A Wolfram Web Resource.
                      http://mathworld.wolfram.com/EuclidsTheorems.html

--- end quoting the website page
http://mathworld.wolfram.com/EuclidsTheorems.html ---

The above is a sloppy and invalid proof attempt of Euclid's Infinitude
of Primes. The major error is that N is necessarily prime and being
prime generates the contradiction. Whenever a proof attempt starts
fishing for a "prime factor" in Euclid's indirect method only signifys
that the mind of the person proving this theorem is too weak, too sloppy
and cannot hold to pristine clear and correct logic.

Here is both the Indirect and Direct proofs of Euclid's Infinitude of
Primes which I posted to the Internet as long ago as 1993-1994. And
MathWorld is seen as authoritative and educational, yet they seem to not
mind keeping a bogus proof for over 10 years on the Internet.

####
** posted 1993-1994 proofs of Infinitude of Primes by Archimedes
Plutonium **
       Valid proofs of Euclid's Infinitude of Primes
Here is the valid Direct proof of IP
 Infinitude of Primes Proof, *DIRECT Method*

(1) Statement: Given any finite collection of primes 2,3,...,pn
possessing a
cardinality n Reason: given

(2) Statement: we find another prime by considering W+1 = (2x3x...xpn)
+1
Reason: can always operate on given numbers

(3) Statement: Either W+1 itself is a prime
Reason: numbers are either unit, composite or prime

(4) Statement: or else it has a prime factor not equal 2,3,...,pn
Reason: numbers are either unit, composite or prime

(5) Statement: If p is not prime, we find that prime factor
Reason: Unique Prime Factorization theorem

(6) Statement: Thus the cardinality of every finite set can be increased

Reason: from steps (3) through (5)

(7) Statement: Since all/any finite cardinality set can be increased by
1 more therefore the set of primes is an infinite set.
Reason: going from the existential logical quantifier to the
universal quantification

###

Here is the valid Indirect proof of IP. (Using Hardy's terminology)
 Infinitude of Primes Proof, *INDIRECT Method*

(1) The prime numbers are the numbers 2,3,...,pn,... of set S
Reason: definition of primes

(2) Suppose finite, then 2,3,...,pn is the complete series set
Reason: supposition step

(3) Set S are the only primes that exist
Reason: from step (2)

 Note: This step follows immediately from the "Suppose primes are
finite".
And it is a very important step and because it is so much like the
statement
"Suppose the set S of primes is finite", for it is this statement
"The set S is all the primes that exist".
And it is this step that disallows any other prime factor search once
W+1
is
formed. W+1 is your only allowable prime candidate in the indirect
method,
otherwise you are self-contradicting your
own logic. The math logic is supposed to get you the contradiction, not
you, in your own illogic.)

(4) Form W+1 = (2x3x,...,xpn) + 1
Reason: can always operate and form a new number

 Inspection steps for there are three possibilities

(5) Is W+1 unit? no
 Reason: it is 2 or greater by the product

 Premissa steps (6) through (8)

(6) Is W+1 prime? yes
 Reason: Unique Prime Factorization theorem combined with the
 definition of what prime is.

(7) Is W+1 composite? yes
 Reason: step (2) and (4) that any number outside of Set of all existing

 primes is a composite number

(8) Contradiction
Reason: theorem that no number can be both prime and composite
simultaneously

Note: another theorem that has been generally neglected in Number
theory.

(9) Reverse supposition step and primes are infinite
Reason: steps (1) through (8)

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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