| Julio (and invited eavesdroppers),
|
| I've just read your July 9
January 9
| posting on "25 = 27 - 2", which contains preliminary translations of
| Articles 188 and 195 of Euler's "Algebra", book II, chapter 12.
:
: Any comments? ...
:
| Well, first a minor matter: On two occasions in your posting, you write
|
| [Gauss's original follows:]
|
| I assume you meant "Euler", not "Gauss".
Yes, of course. Sorry!
| But my overall comment is that it doesn't look like Euler is offering his
| analysis as any kind of proof. It's more in the spirit of "Here's a good
| way of generating solutions to problems like this", with a remark that
| "Sometimes there are solutions that don't arise from this procedure, and
| it'd be important to understand where they come from".
|
| So, if this is all that Euler wrote on the subject, how did he ever get
| falsely credited with having proved (or having claimed to prove) Fermat's
| assertion?
Dear Jim,
I am almost certain that this is ALL that Euler wrote on the subject.
I'd love to be proved wrong! So, I am suggesting right now to forget
about what others might have ever claimed on this issue. What I am
quoting is the transcription (and translation!) of primary sources.
In September 1875, Pepin published in the _journal de Liouville_ a long
essay entitled "Sur certains nombres complexes compris dans la formule
a + b R(-c)". I have quickly translated the *introduction* of that paper.
His remarks are helpful enough, I think, to have an overall picture of
what was known and thought at the time.
I hope the following prove useful and informative.
1. Euler, in the second Part of his Algebra (Chap. XII, XIII and
XIV), solves several problems of indeterminate Analysis using
complex numbers of the form p+qR(c), where $p$ and $q$ denote two
as-yet-undetermined whole numbers, and $c$ either represents a
negative integer, or a positive nonsquare integer. Having recognized
that the product of two similar functions
p + cq^2, r^2 + cs^2
is a similar function
(pr +- cqs)^2 + c(ps -+ qr)^2,
he concludes that, to transform the formula x^2 + cy^2 into a square,
when $c$ is a prime, it is necessary to put
x + yR(-c) = m(p + qR(-c))^2,
what becomes
x = m(p^2 - cq^2), y = 2mpq,
where $m$ denotes the highest common factor of the two numbers $x$
and $y$. Whenever $c$ can be decomposed into two factors $a$ and $b$,
Euler writes
x + yR(-a.b) = (R(a).p + R(-b).q)^2,
thence
x = ap^2 - bq^2, y = 2pq, x^2 + cy^2 = (ap^2 + bq^2)^2
2. It is obvious that the values of $x$ and $y$, obtained by this
method, make the formula x^2 + cy^2 a square. But when this method
does not render any solution, it would be necessary to take care
of not affirming that the problem is impossible. Indeed, when this
method gives all the possible solutions of the problem, it is fine;
but yet it is often insufficient. Thus, it is necessary to introduce
these complex numbers into Analysis only by finding the necessary
conditions so that their use be legitimate. This is what Euler did
not do. For example, he poses the problem: "To find the squares
which, multiplied by 5 and added to 7, produce cubes." This question
obviously amounts to finding, amongst all the solutions of the
equation 5x^2 + 7y^2 = z^3, those in which y=+1 (or -1). However,
by applying his method to express the solutions of this equation,
Euler obtains the following formulae:
x = 5p^3 - 21pq^2, y = q(15p^2 - 7q^2), z = 5p^2 + 7q^2,
in which we have to attribute all the rational values to the yet-
undetermined rational numbers $p$ and $q$ such that $x$, $y$, and $z$
be whole numbers. Next, having recognized that the only solution
for which $y$ equals +1 (or -1) corresponds to the values p=q=+1/2
(or both -1/2), that render x=+2 (or -2), y=+1 (or -1), Euler concludes
that 4 is the only square which answers the question.
This conclusion is not legitimate, since the formulae given by Euler
are not the only ones that can transform 5x^2+7x^2 into a cube. This
condition is also satisfied by the following formulae:
x = 2p^3 + 9p^2q - 18pq^2 - 16q^3,
y = p^3 - 9p^2q - 18pq^2 + 8q^3,
z = 3p^2 + 2pq + 12q^2.
We may deduce Euler's solution writing
q=0, p=1, and so y=1, x=2, z=3.
These formulae, and those of Euler, contain all the solutions of
the problem: we even obtain them all by giving only whole values
to the numbers $p$ and $q$. However, to conclude that 4 is the
only square which, multiplied by 5 and added to 7, makes a cube,
it remains to prove that the only solution (in integers) of the
equation
p^3 - 9p^2q - 18pq^2 + 8q^3 = +1 (or -1)
is obtained when plugging q=0, p=+-1 back into the equation. This
is what we will do in order to complete Euler's demonstration.
We may write:
(p+2q)(p^2 - 11pq + 4q^2) = p^3 - 9 p^2q - 18pq^2 + 8q^3.
This formula can thus be reduced to +1 (or -1) only for values of
$p$ and $q$ suitable to check the two equations
p + 2q = +1 (or -1), p^2 - 11pq + 4q^2 = +1 (or -1).
However, the system of these two equations does not admit solutions
other than the rationals q=0, p=+1 (or -1); p=0, q=+1/2 (or -1/2),
or nonrational solutions. The two rational solutions give x=+2
(or -2), y=+1 (or -1), z=3; what verifies Euler's assertion.
3. We cannot regard any of the two Fermat's theorems as entirely
proven, which is the object of the first two questions of Chapter
XII, as long as we have not justified the use of the preceding
method for the two formulae x^2 + y^2, x^2 + 2y^2. But, this is
what was neither worked out by Euler nor by Legendre. On this issue,
Legendre contented himself with quoting Euler's demonstrations.
Gauss is the first who introduced the complex numbers in a fully
rigorous way; but he did it only for the complex numbers a+bR(-1),
of which he gave the theory in his second Memoir _On the biquadratic
residues_. We also find the substance of this theory in Dirichlet's
Memoir _On the quadratic form with coefficients and indeterminate
complexes (Journal of Crelle, t. XXIV)_. The same scientist has
considered the complex numbers a+bR(5) in his Memoir _On the
impossibility of the equation x^5+y^5+z^5=0,_ and the complex
numbers of form a+bR(-7) in his Memoir _On the impossibility of
the equation x^14 + y^14 = z^14_. In the first of these Memoirs
(_Journal of Crelle_, t. III, p. 354) Dirichlet shows that the
only way to check the equation
P^2 - 5Q^2 = z^5
when the second unknown Q is divisible by 5, is to write:
P + QR(5) = (\phi +- \psi*R(5))^5, z = (\phi)^2 - 5(\psi)^2;
and in the other Memoir (_Journal of Crelle_, t. IX, p. 391), he
bases upon this theorem, that the most general way to raise the
formula (\phi)^6 + 7(\psi)^2 to the fourteenth power, where \psi
and \phi denote two coprime integers, is to put
(\psi)^3 + (\psi)R(-7) = ( g + hR(-7) )^14,
and he indicates for this theorem a demonstration similar to that
of the equation
P^2 - 5Q^2 = z^5.
4. Are the complex numbers a+bR(-1), a+bR(5), a+bR(-7), the only
ones for which the Euler's method can be applied? We will show that
there are several others. Moreover, the use of the complex numbers
a+bR(-7) is not limited to the particular case considered by
Dirichlet; it is as extended, we will see it later, as that of the
complex numbers a+bR(-1), at least when we deal with the issue of
equating the formula x^2+7y^2 with a power of an odd number. Then,
which are the complex numbers that we can introduce into the
indeterminate analysis of the integers, and which are the necessary
conditions to legitimate their usage? Such is the question that I
propose to solve in the first Part of this Memoir; admittedly in an
incomplete way, limiting myself to the complex numbers of the form
a+bR(-c), where $a$ and $b$ denote any integers, and $c$ a positive
integer. The second Part of the Memoir is devoted to show, solving a
great number of problems, the advantages of the theory presented in
the first Part.
* * * * * * * * * * * * * * * * * * * * * * *
[Pepin's original *introduction* follows:]
1. Euler, dans la seconde Partie de son Alge\bre (Chap.
XII, XIII et XIV), re/sout plusieurs questions d'Analyse
inde/termine/e a\ l'aide des nombres complexes compris
dans la formule ge/ne/rale p+qR(c), dans laquelle $p$
et $q$ de/signent deux nombres entiers quelconques, et
$c$ un nombre entier ne/gatif, ou un nombre entier
positif non carre/. Ayant reconnu que le produit de deux
fonctions semblables
p + cq^2, r^2 + cs^2
est une fonction semblable
(pr +- cqs)^2 + c(ps -+ qr)^2,
il conclut que, pour transformer en carre/ la forme
x^2 + cy^2, dans le cas ou\ $c$ est premier, il faut
poser
x + yR(-c) = m( p + qR(-c) )^2,
ce qui donne
x = m(p^2 - cq^2), y = 2mpq,
$m$ de/signant le plus grand commun diviseur des deux
nombres $x$ et $y$. Dans le cas ou\ $c$ peut se
de/composer en deux facteurs $a$ et b, Euler pose
x + yR(-a.b) = ( R(a).p + R(-b).q )^2 .
2. Il est e/vident que les valeurs de $x$ et de $y$,
obtenues par cette me/thode, rendent la formule
x^2 + cy^2 e/gale a\ un carre/; mais il faudrait bien
se garder d'affirmer qu'un proble\me est impossible
lorsque cette me/thode ne donne aucune solution. Il
est impossible lorsque cette me/thode ne donne aucune
solution. Il est des cas, il es vrai, ou\ ce moyen
donne toutes les solutions possibles du proble\me;
mais plus souvent encore il est insuffisant. Il est
donc ne/cessaire de n'introduire ces nombres complexes
dans l'Analyse qu'en de/terminant les conditions
ne/cessaires pour rendre leur emploi le/gitime. C'est
ce qu'Euler n'a pas fait; par exemple, il se propose
cette question: "Trouver les carre/s qui, multiplie/s
par 5 et ajoute/s a\ 7, produisent des cubes." Cette
question revient e/videmment a\ trouver, parmi les
solutions entie\res de l'e/quation 5x^2 + 7y^2 = z^3,
celles dans lesquelles y=+1 (or -1). Or, en appliquant
sa me/thode, Euler obtient, pour exprimer les solutions
de cette e/quation, les formules suivantes:
x=5p^3 - 21pq^2, y=q(15p^2 - 7q^2), z=5p^2 + 7q^2,
dans lesquelles on doit attribuer aux inde/termine/es
$p$ et $q$ toutes les valeurs rationnelles d'ou\
re/sultent des valeurs entie\res pour x,y et z; puis,
ayant reconnu que la seule solution dans laquelle $y$
soit e/gal a\ +-1 correspond aux valeurs p=q=+-1/2,
qui donnent x=+-2, y=+-1, Euler conclut que 4 est le
seul carre/ qui re/ponde a\ la question.
Cette conclusion n'est pas le/gitime, puisque les
formules donne/es par Euler ne sont pas le seules qui
puissent transformer en un cube la forme 5x^2+7x^2;
on satisfait e/galement a\ cette condition par les
formules suivantes:
x = 2p^3 + 9p^2q - 18pq^2 - 16q^3,
y = p^3 - 9p^2q - 18pq^2 + 8q^3,
z = 3p^2 + 2pq + 12q^2.
On en de/duit la solution d'Euler en posant
q=0, p=1, d'ou\ y=1, x=2, z=3.
Ces formules et celles d'Euler renferment bien toutes
les solutions du proble\me: on les obtient me^me toutes
en ne donnant que des valeurs entie\res aux nombres $p$
et $q$; mais, pour conclure que 4 est le seul carre/
qui, multiplie/ par 5 et ajoute/ a\ 7, produit un cube,
il reste a\ de/montrer que l'unique manie\re de verifiquer
en nombres entiers l'e/quation
p^3 - 9p^2q - 18pq^2 + 8q^3 = +1 (or -1)
est de poser q=0, p=+-1. C'est ce que nous allons faire,
afin de comple/ter la demonstration d'Euler.
On a identiquement
(p+2q)(p^2 - 11pq + 4q^2) = p^3 - 9 p^2q - 18pq^2 + 8q^3;
cette formule ne peut donc se re/duire a\ +-1 que pour des
valeurs de $p$ et $q$ propres a\ ve/rifier les deux e/quations
p+ 2q = +-1, p^2 - 11pq + 4q^2 = +-1.
Or le syste\me de ces deux e/quations n'admet que les
solutions rationnelles q=0, p=+-1; p=0, q=+-1/2, et des
solutions non rationnelles. Les deux solutions
rationnelles donnent x=+-2, y=+-1, z=3; ce qui ve/rifie
l'assertion d'Euler.
3. On ne peut pas non plus conside/rer comme entie\rement
de/montre/s les deux the/ore\mes de Fermat, qui son l'object
des deux premie\res questions du Chapitre XII, tant que
l'on n'a pas justifie/ l'emploi de la me/thode pre/ce/dente
pour les deux formes x^2 + y^2, x^2 + 2y^2. Or, c'est ce
qui n'a e/te/ fait ni par Euler, ni par Legendre; celui-ci
s'est contente/ de reproduire sur ce point les
de/monstrations d'Euler. Gauss est le premier qui ait
introduit les nombres complexes d'une manie\re
comple/tement rigoureuse; mais il ne l'a fait que pour les
nombres complexes a+bR(-1), dont il a donne/ la the/orie
dans son second Me/moire _Sur les re/sidus biquadratiques_.
On trouve aussi les e/le/ments de cette the/orie dans un
Me/moire de Dirichlet _Sur les formes quadratiques a\
coefficients et a\ indetermine/es complexes (Journal de
Crelle, t. XXIV)_. Le me^me savant a conside/re/ les nombres
complexes a+bR(5), dans son Me/moire _Sur l'impossibilite/
de l'e/quation x^5+y^5+z^5=0_ et les nombres complexes de
la forme a+bR(-7) dans son Me/moire _Sur l'impossibilite/
de l'e/quation x^14 + y^14= z^14. Dans le premier de ces
Me/moires (_Journal de Crelle_, t. III, p.354) Dirichlet
de/montre que l'unique manie\re de ve/rifier l'e/quation
P^2 - 5Q^2 = z^5
quand la seconde inde/termine/e Q doit e^tre divisible par
5, est de poser
P+ QR(5) = ( phi +- psi*R(5) )^5, z = (phi)^2 - 5(psi)^2;
et dans l'autre Me/moire (_Journal de Crelle_, t. IX, p.
391) il s'appuie sur ce the/ore\me, que la manie\re la
plus ge/ne/rale de rendre e/gale a\ une quatorzie\me
puissance la formule (phi)^6 + 7(psi)^2, ou\ psi et phi
de/signent deux nombres entiers et premiers entre eux,
est de poser
(psi)^3 + (psi)R(-7) = ( g+hR(-7) )^14,
et il indique pour ce the/ore\me une de/monstration
semblable a\ celle qui concerne l'e/quation
P^2 - 5Q^2 = z^5.
4. Les nombres complexes a + bR(-1), a+bR(5), a+bR(-7)
sont-ils les seuls auxquels on puisse appliquer la
me/thode d'Euler? Nous montrerons qu'il en est plusieurs
autres. De plus, l'emploi des nombres complexes a+bR(-7)
n'est pas borne/ au cas particulier conside/re/ par
Dirichlet; il est aussi e/tendu, nous le verrons, que
celui des nombres complexes a+bR(-1), du moins tant qu'il
s'agit d'egaler la formule x^2+7y^2 a\ une puissance
d'un nombre impair. Quels sont donc les nombres complexes
que l'on peut introduire dans l'analyse inde/termine/e
des nombres entiers, et quelles sont les conditions
ne/cessaires pour en le/gitimer l'usage? Telle es la
question que je me propose de re/soudre dans la premie\re
Partie de ce Me/moire; non pas, il est vrai, d'une manie\re
comple\te, mais en me bornant aux nombres complexes compris
dans la formule a+bR(-c), ou\ $a$ et $b$ de/signent des
nombres entiers quelconques, et $c$ un nombre entier et
positif. La seconde Partie du Me/moire est consacre/e a\
montrer, par la solution d'un grand nombre de proble\mes,
les avantages de la the/orie expose/e dans la premie\re
Partie.
* * * * * * * * * * * * * * * * * * * * * * *
With best regards from sleepy Montevideo,
Julio Gonzalez Cabillon