MATHEMATICS: D. N. LEHMER 
113 
fll = anAn-m + + • • • + It = 1, 2, . . . 
bi = GnBn-m + ^nB^-m+l + • • • ^m^«-1 + InBn Aq = A -.i = ^_2 = 
• . = A-m+1 = 0, 
ll = dfiLn—m H~ bfiLn—m+l ^w-^w— 1 "h ^n-^n A—rn — Ij CtC. 
Now, as we have seen, in the last set a„, c„, ^„ are all zero, and In 
is equal to unity. This gives the theorem. 
This theorem throws into our hands a straightforward method of writing 
down a determinant equal to unity whose bottom row is any given set of 
positive or negative non-zero integers, with greatest common divisor unity. 
Let this last row be A^^ -^w? . . . K^, L^, and let the co-factors of these 
elements in the last determinant be A^, B'^, C'n, . . . L'n. Then these 
co-factors furnish a set of values for the unknown in the indeterminate equa- 
tion AnX -\- Bnj + . . . LnV = 1, and by multiplying these values through by 
r we get a solution of the equation when the right member is equal to r. 
Moreover since A„A'^ + B^B'p + C„C; + . . . K^K'p + L^L'p = 0 for 
p = n— l,n— 2. . .w— w+lwe may write for the most general value 
of the variables: 
X= rAn-{- sA'n-l + tAn-2 + . . . + 
y = rB'n + sB'n-l + tB'n-2 + pB'n-m+l + qB'n-m 
Z = rCn-^ sCn-1 + tCn-2 + -\-pCn-m+l + qCn-m 
That this is the most general form of the solution follows from the fact since 
the determinant of the co-factors is unity a set of integer values of r, s, t, . . . 
p, q can always be found for any given set of values of x, y, z, . . . 
We give as a numerical example the problem of determining the most gen- 
eral solution of the indeterminate equation: 
33x + 55y + 79s - 99w = r. 
The following is a convenient arrangement of the work of computing the sets 
di, bi, Ci, di, ai, ^i, 7i, etc. 
bn 
Cfi 
7n 
33 
55 
79 
-99 
1 
2 
-4 
22 
13 
33 
33 
0 
1 
1 
13 
11 
11 
22 
0 
0 
1 
11 
11 
9 
13 
0 
0 
1 
11 
9 
2 
11 
0 
0 
0 
9 
2 
11 
11 
0 
1 
1 
2 
2 
2 
9 
0 
0 
4 
2 
" 2 
1 
2 
0 
0 
0 
2 
1 
2 
2 
0 
0 
0 
1 
2 
2 
2 
2 
2 
2 
0 
0 
0 
1 
