CONGRUENCE 101

Then, since 7, = 7,2 — qu7a, § 91.
"= (-2 — qulaa)a + (Mmp_2 — qumn_1)b

Or 7, = lha + mud

Thus we have proved that, if two particular successiv num-
bers, 7,—s and 7,, in the series of 7’s, 7,_; not being the last
in the series, can be written in the desired form, so can the
next one, 7.

Therfor by mathematical induction they can all be written
in this form.

That is, 7, = lIya + mb for all values of 2 from — 1 to #.

240. Th. Having given two integers, a and b, of which
b—= 0, these numbers and all the other numbers in the series
7—1, 70, 71,72, 73, * * *, 7, Of § 226 can be exprest in the form la + mb,
where | and m are integers.

241. The formulas b = s — qulr
and mr = Mrg—2 — QMr—1

show how, having given I_y, Iy, m_y, mo, and the series of ¢’s,
we may find all the I’s and m’s.

We will illustrate by an example, taking ¢ = — 23 and
b = 17, putting our work in tabular form, as below.

 

 

7 q l m check
— 23 1 0 1(—23) +0(17) = — 23
17 0 1 0(—23)+1(17) = 17
— 6|—1 1 1 1(—23)4+1(17)=— 6
- 1|-3 3 4 3(—23)4+4(17) =— 1

0] 6|~171=231(=17(- B+ (- =0

 

In this table the first two columns are filled out as in the
examples of §§ 226-230.

The first two numbers in the / column are always 1 and 0,
those in the m column always 0 and 1.