Questions in higher arithmetic lead frequently to singular phenomena, much more so than in analysis, and this contributes a great deal to their allure. In analytical investigations it is evidently impossible to discover new truths, unless the way to them has been revealed by our mastery of their underlying principles. On the other hand, in arithmetic it is very often the case that, through induction and by some unexpected fortune, the most elegant new truths spring up, the demonstrations of which are so deeply hidden and shrouded in so much darkness, that they elude all efforts, and deny access to the keenest investigations. Furthermore, there are so many surprising connections between arithmetic truths, which are at first sight most heterogeneous, that we not infrequently arrive at a demonstration much desired and sought after through long meditations, by a path very different from that which had been expected, while we are looking for something quite different. Generally speaking, truths of this kind are of such a nature that they can be approached by several very different paths, and it is not always the shortest paths that present themselves at first. With such a truth, which has been demonstrated through the most abstruse detours, it is certainly valuable if one happens to discover a simpler and more genuine explanation.
Among the questions mentioned in the preceding article, a prominent place is held by the theorem containing almost all the theory of quadratic residues, which in Disquisitiones Arithmeticae (Sect. IV) is distinguished by the name fundamental theorem. Legendre is undoubtedly to be regarded as the first discoverer of this most elegant theorem, although the great geometers Euler and Lagrange had long before discovered several of its special cases by induction. I will not dwell here on enumerating the efforts of these men to find a demonstration; the reader is referred to their extensive work which has just been mentioned. However it is permissible to add, in confirmation of what has been stated in the previous article, an account of my own efforts. I had fallen upon the theorem on my own in 1795, at a time when I was completely ignorant of all that had already been discovered in higher arithmetic, and was completely shut out from literary resources. For a whole year it tortured me, and eluded me despite my most strenuous efforts, until at last I received the demonstration that I have delivered in the fourth Section of the aforementioned work. Afterwards, three others presented themselves to me, based on entirely different principles, one of which I delivered in the fifth Section. But all these demonstrations, even if they seem to leave nothing to be desired with regard to rigor, are derived from very heterogeneous principles, except perhaps the first, which nevertheless proceeded by more laborious reasoning, and was burdened by more extensive operations. Therefore, I have no doubt that until now a genuine demonstration has not been given; let it now be up to the experts to judge whether that which has lately been successfully discovered, and which the following pages present, deserves to be decorated with this name.
Theorem. Let
be a positive prime number, and let
any integer not divisible by
the complex of numbers
the complex of numbers
![{\textstyle {\frac {1}{2}}(p+5)\ldots p-1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/afc4170d4db892db05520352047da840aba18077)
Let us consider the minimal positive residues modulo
of the products of
with each of the numbers in
These will obviously all be different, with some belonging to
and others to
Now if it is assumed that, among the resulting residues,
of them belong to
then
will either be a quadratic residue or a quadratic non-residue modulo
according as
is even or odd.
Proof. Let the residues belonging to
be
and let the remaining residues belonging to
be
It is clear that the complements of the latter,
are all distinct from the numbers
and that, taken together, they complete the complex
We therefore have
Now the latter product clearly becomes
Hence we have
or
according as
is even or odd, from which our theorem immediately follows.
The following considerations will be greatly shortened by the introduction of certain notation. We therefore let the symbol
denote the multitude of residues of the products
whose minimal positive residues exceed
Moreover, for any non-integral quantity
we denote by
the greatest integer less than or equal to
so that
is always a positive quantity between
and
We can then easily derive the following relations:
- I.
![{\textstyle [x]+[-x]=-1.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ee2cac65ba209a28832e9b358b4d2e22c5b93c3)
- II.
whenever
is an integer.
- III.
![{\textstyle [x]+[h-x]=h-1.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/abcb533e6915fad7470ac72d6b35c18a038b8b28)
- IV. If
is a fraction smaller than
then
if it is greater than
then ![{\textstyle [2x]-2[x]=1.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a118b40a8dbec93698b3102cbbe99a9eaebe2791)
- V. If the minimal positive residue of an integer
exceeds
modulo
then
if it is less than or equal to
modulo
then ![{\textstyle \left[{\frac {2h}{p}}\right]-2\left[{\frac {h}{p}}\right]=1.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a7fc73f495bb1d7f65bc17c7f54b0513da61b95c)
- VI. From this it immediately follows that
![{\textstyle (k,p)=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aecb8f8ef2a782f0dfe65361d79258735d3900a4)
- VII. From VI and I, we can easily deduce
- Hence, it follows that
has the same or opposite relation to
(insofar as it is a quadratic residue or non-residue) as does
depending on whether
is of the form
or
It is clear that in the former case,
will be a quadratic residue modulo
whereas in the latter case, it will be a quadratic non-residue.
- VIII. We will transform the formula given in VI as follows. From III, we have
- Applying these substitutions to
terms in the series above, we have
- first, if
is of the form
then
- second, if
is of the form
then
- IX. For the special case
it follows from the formulas given above that
with the sign being taken as
or
depending on whether
is of the form
or
Thus,
is even and
whenever
is of the form
or
on the contrary,
is odd and
whenever
is of the form
or ![{\textstyle 8n+5.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d226f133eb8bfc650ba45f3bedba4652e77308d)
Theorem. Let
be a positive non-integer quantity such that no integer can be found among its multiples
up to
. Letting
it is easily concluded that no integer can be found among the multiples of the reciprocal quantities
up to
Then I say that
Proof. Let
represent the series
Then the terms up to the
term inclusively are clearly all
the terms up to the
term inclusively are all
the terms up to the
term inclusively are all
and so on. Hence we have
Q. E. D.
Theorem. Let
be any odd positive integers that are prime to each other. Then
Proof. Suppose, which is allowed, that
Then
is smaller than
but larger than
so
Hence, it is clear that the current theorem follows immediately from the previous theorem by taking
and therefore
It can be demonstrated in a similar manner that if
is an even number, relatively prime to
then
But we do not dwell on this proposition, which is not necessary for our purposes.
Now, by combining the theorem mentioned above with proposition VIII of article 4, the fundamental theorem immediately follows. Indeed, let
and
be any two distinct positive prime numbers, and let
Then by proposition VIII of article 4, it is clear that
and
are always even. But by the theorem of Article 6, we have
Therefore, when
turns out to be even, which occurs if either both
and
are of the form
or if one of them is of the form
it is necessary that either both
and
are even or both are odd. On the other hand, when
is odd, which happens if both
and
are of the form
it is necessary that one of the numbers
and
is even and the other is odd. In the former case, then, the relation of
to
and the relation of
to
(insofar as one is a quadratic residue or non-residue modulo the other) will be identical, and in the latter case they will be opposite.
Q. E. D.