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

THE THEOREMS OF FERMAT AND WILSON

57

~ 31. APPLICATION OF THE PRECEDING RESULTS TO THE THEORY
OF QUADRATIC RESIDUES
In this section we shall apply the preceding results of this
chapter to the problem of finding the solutions of congruences
of the form
az2+z+T ---o mod u
where a, A3, y, u are integers. These are called quadratic congruences.
The problem of the solution of the quadratic congruence
(I) can be reduced to that of the solution of a simpler form of
congruence as follows: Congruence (I) is evidently equivalent
to the congruence
4a22 z+4a3z +4ay =o mod 4a/E. (I)
But this may be written in the form
(2aZ +-)2_32-4a7 mod 4a/.
Now if we put
2aZ+S — x mod 4ayc (2)
and
P2-4ay7=a, 4a/c=m,
we have
x2a mod m. (3)
We have thus reduced the problem of solving the general congruence (I) to that of solving the binomial congruence (3)
and the linear congruence (2). The solution of the latter may
be effected by means of the results of ~ 30. 'We shall therefore confine ourselves now to a study of congruence (3). We
shall make a further limitation by assuming that a and m
are relatively prime, since it is obvious that the more general
case is readily reducible to this one.
The example
x2= 3 mod 5