336 
History of the Theory of Numbers. 
[Chap. XI 
Erroneous remarks^^ have been made on the g. c. d. of 2"" — 1, 3"" — 1. 
]\I. Lecat^° noted that, if a,j is the 1. c. m. of i and j, the determinant 
loyl was evaluated by L. Gegenbauer,^^ who, however, used a law of multi- 
pHcation of determinants valid only when the factors are both of odd class. 
J. Barinaga®^" proved that, if 5 is prime to iV = nk, the sum of those terms 
of the progression A'', N-\-d, iV+25, . . . , which are between nk and n{k-\-hd) 
and which have with n = mp the g. c. d. p, is ^n<l){n/p){2k-\-hd)h. 
R. P. Willaert®- noted that, if P{n) is a polynomial in n of degree p with 
integral coefficients, f{n)=aA'"'-\-P{n) is divisible by D for every integral 
value of 7} if and only if the difference A''f{0) of the Ath order is di\"isible 
by D for k = 0, 1,. . ., p-\-l. Thus, if p = l, the conditions are that /(O), 
/(l),/(2) be divisible by D. 
*H. Verhagen^^ gave theorems on the g. c. d. and 1. c. m. 
H. H. ]\Iitchell^ determined the number of pairs of residues a, b modulo X 
whose g. c. d. is prime to X, such that ka, kb is regarded as the same pair as 
a, b when k is prime to X, and such that X and ax + by have a given g. c. d. 
W. A. Wijthoff^^ compared the values of the sums 
S (-l)'"-WF{(w, a)}, "s m'F{{m,a)}, s = l, 2, 
m=l m=l 
where {m, a) is the g. c. d. of m, a, while F is any arithmetical function. 
F. G. W. Brown and C. M. Ross^^ wTote h, U, ...,ln for the 1. c. m 
the pau-s Ai, A^; A^, Az; . . . ; A„, Ai, and gi, g^, ■ ■ ., gn for the g 
these pairs, respectively. If L, G are the 1. c, m. and g. c. d. of Ai, 
A„, then 
gm . . .gn = G'', 
. c 
^2, 
of 
d. of 
9i92 
C. de Polignac^^ obtained for the g. c. d 
(a\btJi)={a,by{\,fx).(--\ 
\{a, b) (X, m) 
Sylvester^* and others considered the g. c. d 
9n G' 
(a, 6) of a, 6 results like 
fi \ / b X 
J \(a, 6)' (X, mV 
6)' (X, m)> 
of Z)„ and Z)„+i where D^ 
is the n-rowed determinant whose diagonal elements are 1, 3, 5, 7, . . ., 
and having 1, 2, 3, 4, ... in the line parallel to that diagonal and just above 
it, and units in the parallel just below it, and zeros elsewhere. 
On the g. c. d., see papers 33-88, 215-6, 223 of Ch. V, Cesaro" of Ch. 
X, Cesaro^' ' of Ch. XI, and Kronecker^^ of Ch. XIX. 
"L'interm6diaire des math., 20, 1913, 112, 183-4, 228; 21, 1914, 36-7. 
^''Ibid., 21, 1914, 91-2. 
•'Sitzungs. Ak. Wiss. Wien (Math.), 101, 1892, II a, 425-494. 
""Annaes Sc. Acad. Polyt. do Porto, 8, 1913, 248-253. 
^Mathesis, (4), 4, 1914, 57. 
"Nieuw Tijdschria voor Wiskunde, 2, 1915, 143-9. 
"Annals of Math., (2), 18, 1917, 121-5. 
"Wiskundige Opgaven, 12, 1917, 249-251. 
"Math. Quest, and Solutions, 5, 1918, 17-18. 
«'Xouv. Corresp. Math., 4, 1878, 181-3. 
6»Math. Que.st. Educ. Times, 36, 1881, 97-8; correction, 117-8. 
