MATHEMATICS: D. N. LEHMER 
111 
THE GENERAL SOLUTION OF THE INDETERMINATE 
EQUATION: Ax By Cz + = r. 
By D. N. Lehmer 
Department of Mathematics, University of California 
Communicated by E. H. Moore, January 30, 1919 
Little has been done since Jacobi (Werke, 6, 355) in connection with the 
solution of the general linear indeterminate equation. Jacobi has given no 
less than four methods all involving the reduction of the equation by means 
of a set of auxiliary equations to the solution of an equation in two variables 
the solution of which is immediately obtained from the theory of the ord nary 
continued fraction. The solution here presented treats the general equation 
in the same non-tentative way that is found in the continued fraction solution 
for two variables. The method applies equally well when the right hand mem- 
ber is zero and gives a perfectly general solution from which all other special 
solutions may be obtained. 
Consider the set of positive or negative, non-zero integers ai, bi, Ci, . . ., 
ki, h and let m be the number of terms in the set. We derive from this first 
set a second set 02, 62, C2, . . . ^2, h by means of the equations: 
^^2 = ^1 — 0:1^1, 
h= Ci — (3iai 
k2 = h — Kiai, 
h = ai 
where if ai is different from unity the numbers ai, . . . ki are so taken 
that 02, 62, C2, . . . ^2 are the smallest positive residues, not zero, of the num- 
bers 61, ci, . . . ^1, li respectively with respect to the modulus ai. If ai is 
unity then ai, jSi . . . ki are taken equal to 61, Ci, . . . h respectively, so that 
in this case the second set consists of zeros with the exception of the last 
term, h which is unity. 
We derive in the same way a third set from the second by means of the 
equations: 
= b2 — oL^a^ 
bz = C2 — ^2(l2 
kz = h — K2a2 
h = (h 
As in the preceding set, if ^2 is different from unity, we take the numbers 
0L2, ^2, ' ' ' K2 so that as, bz, . . . kz are the smallest positive, non-zero residues 
of &2, C2, . . . h respectively, modulo a2- If a2 is unity, a2, ^2, - • - K2 are taken 
equal respectively to ^2, C2, . . . h, and in this case the third set consists of 
zeros with the exception of the last, k, which is unity. 
