108 SPECIAL TOPICS IN THEORETICAL ARITHMETIC

Also the formulas I = (— 1)¥ s, and m; = (— 1)*;, or
their equivalents s; = (— 1)*, and # = (— 1)*m;, show
that, if 2 is odd, si is the same as /; and ¢ is the opposit of
my, and if % is even, sj is the opposit of /; and #; is the same as m;.

Ex. 1.

 
 

check

 
 
 
  

 

    
   

Hed = 0D =—28
17 o} 1 STy - e 17

      
 

 

=t i =1 S U = — 6
S seE T—Y A(1D) =~ 1
=R CIN—23)- @In= o

   

 

 

 

Ex. 2.
check

 

1'X23 — 0 X 17 =23
023 1 X 17 =17

 

1:X23 — 1 X 17 =
ZoSI3 =S =
3 X23—-4.X17 =
—17 X 23+ 23 X 17 =

 

256. The case of the preceding investigation for which «a is
positiv or zero and b is positiv is especially interesting. For
that case we have the following theorems.

Th. 1If ais a positiv or zero integer, b a positiv integer, if r,
where — 2 < k <n + 1, is as in § 226, if qi is the exact or
lower approximate quotient obtaind by dividing ri_s by i,
where 0 < k < n + 1, and s, and ty, are as in § 254; then

qr 15 positiv except when a < b and k = 1,

Sk 1S positiv except when k = 0,

by 15 positiv except when k = — 1 and when k = 1 and a < by
wn the excepted cases qi, sk, and t, are zero; in all cases when they
have sense qr, sk, and ty, are positiv or zero, g, when 0 < k < n + 1,
sy and t, when — 2 < k < n + 1.