336 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xi Erroneous remarks59 have been made on the g. c. d. of 2Z — 1, 3*—1. M. Lecat60 noted that, if a# is the 1. c. m. of i and j, the determinant |o#| was evaluated by L. Gegenbauer,61 who, however, used a law of multiplication of determinants valid only when the factors are both of odd class. J. Barinaga61a proved that, if 5 is prime to N=nk, the sum of those terms of the progression N, N+5, N+2d, ..., which are between nk and n(k+h$) and which have with n = mp the g. c. d. p, is Jn^(n/p) (2k+h8)h. R. P. Willaert62 noted that, if P(ri) is a polynomial in n of degree p with integral coefficients, f(n)=aAan+P(n) is divisible by D for every integral value of n if and only if the difference A*/(0) of the &th order is divisible by D for fc = 0, 1,..., p+1. Thus, if p = l, the conditions are that /(O), /(I), /(2) be divisible by D. *H. Verhagen63 gave theorems on the g. c. d. and 1. c. m. H. H. Mitchell64 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. Wijthoff65 compared the values of the sums 06 (o6-l)/2 S (-1)—VF{(i»,o)}, S m'F{(m,a)}, «-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. Ross66 wrote Zi, Z2, ..., ln for the 1. c. m. of the pairs Ai, A2; A2, A3; ...; Aw, AI, and #1, gr2, ..., gn for the g. c. d. of these pairs, respectively. If L, G are the 1. c. m. and g. c. d. of AI, A**, ..., An, then C. de Polignac67 obtained for the g. c. d. (a, b) of a, b results like ( \ / T \ \ 7—JA> TT^)- \7—£V> 7^-----\1-(a, 6) (X, /i)/ \ (a, 6) (X, jit)/ Sylvester68 and others considered the g. c. d. of Dn and Z>n+i where Dn 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, Cesaro61 of Ch. X, Cesaro8'9 of Ch. XI, and Kronecker30 of Ch. XIX. wL'intermddiaire des math., 20, 1913, 112, 183-4, 228; 21, 1914, 36-7. «»Ibid., 21, 1914, 91-2. 81Sitzungs. Ak. Wiss. Wien (Math.), 101, 1892, II a, 425-494. 61aAnnaes Sc. Acad. Polyt. do Porto, 8, 1913, 248-253. 62Mathesis, (4), 4, 1914, 57. 63Nieuw Tijdschrift voor Wiskunde, 2, 1915, 143-9. "Annals of Math., (2), 18, 1917, 121-5. KWiskundige Opgaven, 12, 1917, 249-251.