MATHEMATICAL RECREATIONS 205

First we will show that if the equations of § 488 are satisfied
by any values x, y1, Vs, Vs, =+, Yn, 3 and m stands for — n, then
the numbers

I+y,1+y,14ys--+,1+9y,

Sform a geometric progression whose common ratio is (n — 1)/n,
or (m + 1)/m;

also z2=0 (modn — I)
1+ Yo = m* (mod m**1), where 0 = b =pn — 1
¥ — 1= —m" — n (mod mm+l)

and there exists an integer q such that
x=qm"*tl —mr —p+ 1
Ye=m"*m~+ DY gn+ 1) — 1, where 1= k= n
and g=m—D[m+ )Y gn+ 1) — 1] : n;
conversely, for every integral value of q these values of x, v, and
% are integral and satisfy the equations.

First,if 12 k=n—1, ypy = [((m— Dy —1]:m
Hence 1+ yppu=(n— 1)1 + Vi) i 1
and (1 + ye)/(L + 3) = (0 = D/n = (= m — 1)/(— m)
= (m + 1)/m,

which proves the first part of our statement.

Next, since (n — 1)y, = nz, nz>n —1

But, since (n — 1) + 1 = 5, nlln—1 § 198.
Therfor z2>3n —1
And z2=0 (modn — 1)

Next we wish to show that, f 0 = k= 5 — 3.
1 + yur = m* (mod m*+),

that is, that, giving # the values from 0 to # — 1 inclusiv, the
following series of statements are all try :

i