CONGRUENCE 103

Thus dk = lk_lmk —lkmk__1, whereOi kz n
or —1 <k<n+4+1
di—1 = li—emi—y — L_ymy_2, where 1 = k= n+1
AlSO do—llmo'—lom_l—-le—OXO:

dy = Lomy — Limyg
dz = lfl’l”lz — lzml, etc.

243. Th. If di is as defined in § 242, then dp = — diy,
where 1 = k = n, that is, each determinant of two successiv rows
of I's and m’s, after the first of these determinants, is the opposit
of its predecessor.

We have di = lp_1(mi—s — @) — (Ie—2 — Qrbe—1)mr—

= lamp—g — ey = — dr

244. Th. If dy is as defined in § 242, then d, = = 1, accord-
ing as k is even or odd.
For dy = 1 and each d after the first is the opposit of its

predecessor.
Thusd; = — 1,ds = 1,d; = — 1, etc.

245. Th. Ifaand b are two integers, not both zero, two integers
1 and m can be found, such that

loa +mb=a®b
We have a ® b = |ra1| = = 7.1, according as 7,y is
positiv or negativ. §§ 226, 43.
Also Voot = ln1a + Mp_1b § 239.

Therfor, if 7,y is positiv, lnaa + m,b = a Qb
if 7,1 is negativ, — l,1a — m,1b = a Qb
If in the first case we set } = l,—; and m = m,_; and in the

second case ] = — I,_; and m = — m,_;, we have for each
case la + mb = a R b.
For example, taking ¢ = 23 and b = — 17, we find, as in

§ 241, 7,1 = 1, I,y = 3, ma,1 = 4, the quotients being — 1,
— 3, 6.

Hence 3(23) +4(—17) =1 = 23 (— 17).

Taking @ = 24 and b = 45, we find

Yn—a1 = — 3, ln_1 = — 2, Mp—1 = 1