MATHEMATICS: D. N. LEHMER 
215 
To account for these curious theorems, and to place them in their proper 
setting, it was found necessary to study a more general type of continued 
fraction first investigated by Hurwitz. 1 The regular continued fraction for 
the Naperian base was discovered by Roger Cotes 2 to be (2, 1, 2, 1, 1, 4, 1, 1, 
6, 1, 1, 8, 1, 1,. . . .)j or as we may write it (2, 1, 2n, 1, 1), n = 1, 2, 3,. . . 
The proof of the remarkable sequence of partial quotients was first given by 
Euler, 3 by means of the theory of the Riccati equation. Euler established a 
number of other interesting expansions, such as: 
( l,l,4»+l), rc = 0 , 1, 2, 3, ... . 
(l,s(2»+l) - 1,1), n = 0, 1, 2, 3, . . . . 
((4» + 2) j),» = 0, 1, 2, 3, ... . 
The continued fractions studied by Hurwitz may be written in the form; 
(ft, to, q$, • • ■ q r , <Pi (n), cp 2 (n), <p 3 (n), . . . . <p k (») ), n = 0, 1, 2, 3, ... . 
where the 'irregular' partial quotients, qi, q2,...q r are rational, and with 
the possible exception of qi, all positive. The functions <Pi(n) are rational 
integral functions whose degrees may, some or all, be zero. If the functions 
are all of degree zero, the fraction becomes an ordinary periodic continued 
fraction. The fractions discovered by Euler are seen to be Hurwitzian 
fractions where all the functions are of degree zero except one which is of the 
first degree. The general type of such a fraction with no irregular partial 
quotients is 
0*i, a 2, a 3 , .... a k _ u ma k + b), m = 0,'l, 2, . . . . 
and for such fractions I have been able to prove trie congruences: 
A 2n k-i — 1 = B 2nk — 0 (mod n), 
A 2nk ^B 2nk _ 1 ^ (- I)**-* (modn); 
Where A m /B m is the mth convergent to the continued fraction, and n is any 
number prime to A k , A k ^ and B k _ x \ a h a 2 , a s , . . .a k -\ being positive or nega- 
tive integers or zero; A k , A k _ 1} B k _i positive or negative integers not zero. 
The discussion is based on the following two theorems: 
Theorem I. If A m /B m is the mth convergent of (ai, (h, • ■ wflj, 
m = 1, 2, 3,. . . and A' m jB' m is the mth convergent of a k _ 2 , ■ ■ ■ .(h, 
a h — a k m — 2M), m = 1, 2, 3,. . . ; where M = (B k _i + A k __ 2 )/ A k _i, where 
<h ? 02, a 3 . . .a k _ h are positive or negative integers or zero, and where a k , A k -i 
are positive or negative integers, not zero, then: 
y/ e = 
\/ e = 
</e+ 1 
</e- 1 " 
