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[HM] Euclid and the unique factorization theorem
Posted:
Feb 5, 2000 9:35 AM
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In his thoughtful {\it Mathematical reflections} (Amer. Math. Monthly,
1974, 827--852), the "discerning antiquarian" Solomon Bochner
(1899--1982) wrote:
\begin{quotation}
[Leonard Dickson's] reluctance to present the fundamental theorem [of
arithmetic] in an otherwise very conscientious historical account
[Dickson's three-volume {\it History of the Theory of Numbers}
(1919--1923)] may be due to the fact that, while this theorem is
sometimes vaguely attributed to Euclid, it had apparently not been
stated expressly before 1801, when Gauss featured it in his {\it
Disquisitiones arithmeticae}, with some emphasis, and rather early in
the work. The nearest that Euclid himself came to this theorem was his
proposition (vii, 24): "if two numbers be prime to any number, their
product also will be prime to the same."
(...)
Of course, what had discomfited Dickson was not just the fact that the
fundamental theorem does not occur in Euclid, but, undoubtedly much
more so, that none of the great number-theorists in the 17th and 18th
centuries, like Fermat, Wallis, Euler, Lagrange, etc., had chanced
upon it either.
(...)
But there is no substance to assertions that the fundamental theorem
had been consciously known to mathematicians before Gauss, but that
they had neglected to make the fact known.ÃÂ We think that the 17th,
and even the 18th, century was not yet ready for the peculiar kind of
mathematical abstraction which the "fundamental theorem" involves;
just as only the 19th century was comfortably prepared to
conceptualize satisfactorily the notions of real number, limit,
derivative, convergence, etc.
\end{quotation}ÃÂ
According to Bochner, the paragraph quoted by Martin Davis from Hardy
and Wright's famous book "was triggered by remarks made to G. H.
Hardy by myself, and I well remember that the remarks were made,
sometime in 1933, in Hardy's large living room in Trinity College".
Of course, there are some other delicate questions to be taken into
consideration here.
1) Consider, for example, the Greek neglect of higher order
quantifiers. I agree with Karlis Podnicks when he says that "the
natural number concept of Greeks corresponds to first order arithmetic
and that this concept remained unchanged up to 1870s." The relevance
of this remark becomes clear when we note that the "fundamental
theorem" is not a first order statement --- since it involves the
higher notion of a SEQUENCE of primes.
2) Consider also the existence of nonstandard models of arithmetic
which "satisfies all the properties of order, addition, and
multiplication which could reasonably be considered as prior to" the
EXISTENCE part of the fundamental theorem, but in which that existence
fails. (See J. C. Shepherdson's {\it Weak and strong induction}, Amer.
Math. Monthly, 1969, 989--1004, Theorem 2.9, p. 993.)
3) Compare the classical proof of the irrationality of Sqrt[2] with
the more "powerful" one that rests essentially on the fact that
O[m . n, 2] = O[m, 2] + O[n, 2],
where, more generally,
O[m, n] = the greatest natural (0 included) $e$ such that $n^e | m$.
The logarithmic behavior of O with respect to $m$ for PRIME values of
$n$, is due to the fact that, for a prime $p$, if $p | mn$ then $p |
m$ or $p | n$ --- which is crucial to the UNIQUENESS part of the
fundamental theorem.
4) In the light of these facts, can we really say that "Already in
300 B.C., Euclid was aware of the fundamental theorem of arithmetic"?
5) Just as a curiosity: Which fact really deserves to be called "the
fundamental theorem of arithmetic"? According to J. V. Uspensky and
M. A. Heaslet in their book {\it Elementary Number Theory}
(McGraw-Hill, 1939, p. 32), the result which "is rightly called the
fundamental theorem of arithmetic" is this one:
"If the product $ab$ of two integers is divisible by an integer $c$,
and $c$ and $b$ are relatively prime, then $a$ is divisible by $c$."
More generally, we can prove that
IF $c | ab$ THEN $c | a (c,b)$
is true on a integral domain < R, plus, times, 0, 1 > for which
a) GCD[a,b]\neq \emptyset for all $a, b\in R$;
b) Every gcd $(a,b)$ is a linear combination of $a$ and $b$.ÃÂ
If, in addition, $R$ is principal, we can prove that the notions of
"prime" and "irreducible" coincide on $R$.
ÃÂ
Carlos C\'esar de Ara\'ujo from rainy Belo Horizonte.
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