100 SPECIAL TOPICS IN THEORETICAL ARITHMETIC

238. In the proof of the next theorem we will need to use
the principle of mathematical induction in another form as
follows:

Principle of Mathematical Induction. If the first two of a
series of statements are tru, and if, provided any two successiv
statements in the series are tru, so is the next statement, if there
is a next, then the statements are all tru.

This principle is axiomatic. Itis a special case of the follow-
ing, also axiomatic:

n being a primary number, if the first n of a series of statements
are tru, and if, provided any n successiv statements in the series
are tru, so is the next statement, if there is a next, then the state-
ments are all tru.

239. Th. Having given two integers a and b, of which b—= 0,
if 71, is any of the numbers r_y, ro, 11, 72, 73, ***, a Of § 226 and
qr is the quotient used in dividing rir—s by rr— (where 1 = k = n),

if we set li=1 m_y = 0
lo =) Mmoo = 1
Iy = lh—s — gl My = Myr—2 — JrMg—1
then
7, = Lo + mib, if —1=k=n

To prove that 7, = Ixa + m;b for all values of & from — 1
to 7, we have

a=1-a+0-b

and b=0-a+1-b
Hence 71 = l_ia + m_ib

and ro = loa + meb

Therfor the first two of the 7’s have been exprest in the
desired form.

Now suppose we know that two particular successiv 7’s, say
722 and 7,_1, can be written in the desired form, 7,1 not being
the last in the series of 7’s.

That is, suppose that

Theo = lp_2a + mp_2b
and The1 = @ + mp_1b, where 1 = h = n,