Colloquium publications.

INVARIANTS AND NUMBER THEORY.

7

multiplies a particular xi (i < m) by pk and Xm by p-k is of determinant unity.
First, let r < m. Applying transformations of the last type
to (3), we obtain
(6) X12,.. P + + + * + X
Under the transformation of determinant unity
xi = aX + fXj, xtj = - 3Xj + axe, xm = (a2 + 2) —1Xm,
Xi2 + Xj2 becomes (a2 + (2)(Xi2 + Xj2). Choose* integers a, 3
so that
(7) p(a 2 + 32) _ 1 (mod p).
Hence the sum of two terms of (6) with the coefficient p can be
transformed into a sum of two squares. Thus by means of a
linear transformation, with integral coefficients of determinant
unity, qm can be reduced to one of the forms
(8) X12 + * + X-1 + Xr2, X2 + * +X '_ - +pXr2 (O<r<m).
Next, let r = m. We obtain initially
X12 + + X+s2 + PX* +.s + P+ Pi-i1 + 0-Xm2,
in which o- need not equal p as in (6). If there be an even number
of terms with the coefficient p, we obtain as above a form of
type (41). In the contrary case, we get
f = X12 + + 2 + r-2 + PX2-l + p-1DX2.
If D -p21+1 (mod p), f is transformed into (41) by
m-1 = - plXn, Xm = P-Xm-1.
But if D p21, f is reduced to (4k) by the transformation.m-1 = aXm-1 + 6p211Xm, Xm = - 5Xm-l + apXm,
p(a2 + p21-252) -1,
If p =5, p = 2, we may take a = 3 = 2. For any p, either there is an
integer I such that 12 - 1 (mod p) and we may take p(a + I3) = 1,
a - 1/ - 1; or else x2 + 1 takes 1 - (p - 1)/2 incongruent values modulo p,
no one divisible by p, when x ranges over the integers 0, 1, *., p - 1, so
that x2 + 1 takes at least one value of the form p2e-1. In the latter event,
a = p-, 3 = xa satisfy (7).