MATHEMATICS: D. N. LEHMER 
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iWcrf+i) - (~ (X+W) A mk-i (mod »), 
with a similar congruence in the 23 's. 
With these values for the convergents of order 2k\ and 2k\ — 1 it is pos- 
sible to proceed to those of order 2kn and 2kn — 1 with the results as noted 
above 
A 2nk ^B 2nk _ l ^(-l) nk - 1 (mod rc), 
A 2 nk-i= B 2nk =0 (mod n), 
and these last results are, without much difficulty, shown to be valid also for 
the more general fraction 
(a u a 2 , a 3 . . . . a k _ u a k m + b), m = 1, 2, 3, . . . . 
where w is prime to a*, A k -i and 2?£_i. If we take the still more general 
fraction with irregular partial quotients: 
(Vi- Qi, Qz • • • - <7r, #2< &3 - - • » a h m-\-b), m — 1 2 3. 
the values given above for the convergents of order 2nk + r and 2«£ + r — 1 
are no longer correct but must be replaced by the rth and (r — l)st con- 
vergents of the fraction (q h q 2 , qs q r ). 
The above mentioned divisibility theorems concerning the Naperian 
base are easily derived from these general theorems. The special cases 
where n is not prime to 2ak and to A k ^ x and to B k _i have been examined 
in detail. When A k is divisible by n the fraction reduces modulo n to a purely 
periodic fraction of k partial quotients with no relations between them. 
The number of terms in the period of the convergents to such a fraction de- 
pends upon the discriminant D of the quadratic equation of which the fraction 
is one root. For a prime modulus p the period of the convergents is a di- 
visor of p + (D/p) where (D/p) is the symbol of Legendre. 
For the case n a divisor of A k _ x (or B k -i) the period of the convergents 
is a multiple of nke where e is the exponent to which Ak (or B k ) belongs 
modulo n. It is curious that for these special cases the period of the con- 
vergents does not seem to be definitely assignable as in the general case. 
In this investigation we are concerned only with the successive values of 
the numerators and denominators of the convergents, and not at all with 
the existence or non-existence of a limiting value of the convergents. Certain 
of the fractions involved are closely related to those called 'semi-regular' 
whose convergence has been studied by Tietze, 4 and the rules which apply to 
semi-regular continued fractions may be easily modified to apply to them. 
The research here carried out is really an arithmetical study of the series of 
numbers which satisfy certain difference equations and, as Professor BirkhofT 
