The theory of numbers, by Robert D. Carmichael ...

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THEORY OF NUMBERS

every prime of either of the forms 8k + and 8k +7 and a quadratic non-residue of every prime of either of the forms 8k+3
and 8k+5. We may state that result as follows: A given
prime number of either of the forms 8k + i and 8k + 7 is a divisor
of some number of the form X2-2, where x is an integer; no
prime number of either of the forms 8k+3 and 8k+5 is a
divisor of a number of the form x2-2, where x is an integer.
The result just stated is a theorem in a discipline of vast
extent, namely, the theory of quadratic forms. Here a large
number of questions arise among which are the following:
What numbers can be represented in a given form? What is
the character of the divisors of a given form? As a special
case of the first we have the question as to what numbers can
be represented as the sum of three squares. To this category
belong also the following two theorems: Every positive integer
is the sum of four squares of integers; every prime number of
the form 4n+ may be represented (and in only one way) as
the sum of two squares.
For an extended development of the theory of quadratic
forms we refer the reader to Bachmann's Arithmetik der Quadratischen Formen of which the first part has appeared in a
volume of nearly seven hundred pages.
It is clear that one may further extend the theory of arithmetic forms by investigating the properties of those of the third
and higher degrees. Naturally the development of this subject
has not been carried so far as that of quadratic forms; but
there is a considerable number of memoirs devoted to various
parts of this extensive field, and especially to the consideration
of various special forms.
Probably the most interesting of these special forms are the
following:
an+ a -=a n-1+ -+... + n-1
a-3
where a and 3 are relatively prime integers, or, more generally,
where a and f are the roots of the quadratic equation
x2-ux-+v=o where u and v are relatively prime integers. A