36 SPECIAL TOPICS IN THEORETICAL ARITHMETIC

 

 

9 =11+ 5(— 2) + 8(— 2)* + 3(— 2)°

95. We will now suppose that the 7’s of § 92 are infinit in
number, all numerically greater than 1, and that the quotients
are exact or lower or upper approximate quotients. Thus the
quotients will now not be given at pleasure but will be found.

Th. If a is any integer and r, 73, 3, * ++ an infinit series of
integers abl numerically greater than 1, if also qi, where k = 1,
is a quotient, exact, or lower or upper approximate, obtaind by
dividing its predecessor in the series a, qi, @2, qs, *** by ri (for
example, q: is a quotient got by dividing a by r, g2 a quotient
got by dividing q, by rs), if also qo = a; then for some value of n
gn will be numerically less than 7,41

Suppose that all the numbers qo, ¢1, ¢2, g3, ***, Qx, *** are
numerically greater than their divisors, that is,

lgo] > |nl, |l > 72|, |@e| > |7sl, <=, l&] > l7en], -

Then, since |gr—1| > |7x| > 1, where 2 = 1,
|@r—| > |l §§ 88, 90.

Thus [go| > |@i] > [ge| > -+ > |gx| > |7%4a| > 1, where
k is as large as we please.

Then |a| > |gi| > |g| > -+ > |@| > 1

Hence we will have an infinit number of distinct integers
between |a| and 1, which is of course impossible.

Therfor at least one of the ¢'s, say g, must be such that
lgn| = | 7ms1 |

If |gn < |#ms1|, our theorem is tru and #n = m.