32 HISTORY OF THE THEORY OF NUMBERS. [CHAP. 1 



He studied (pp. 20-25) the number 0(a, k) of polygonal divisors of order a 

 of a positive integer k; every integer has in mean two triangular divisors, 

 ?T 2 /6 square divisors, etc. 



A. Goulard and A. Emmerich 189 found two consecutive integers of which 

 one is a square and the other triangular. In x 2 ^y(y + 1) = 1, set 

 2x = z, 2y + 1 = t, whence 2z 2 t 2 = 7 or 9, which are reduced to the 

 Pell equations u 2 2v 2 = 1, or + 1, and solved. 



P. Bachmann 190 gave an excellent exposition of Cauchy's 93 proof of 

 his modification of Fermat's theorem: every integer is a sum of ra ra-gonal 

 numbers of which all but four are or 1. 



R. W. D. Christie 191 noted two formulas of the type 



44 4 



]C A(a t - + n) = A(<r a + n), a = | X) * 

 i=i t=i =i 



J. W. West 192 noted that if A a = 6 A& + 1, A is not a square. 



R. W. D. Christie 193 proved that, if p" is the rath r-gonal number, 



(2n) 3 (r - 2)p7 + K4n 3 - n)y 2 = (x - y) 2 + (x - 3y) 2 + (x - 



x = 2mn(r 2), y = r 4. 



W. A. Whitworth and A. Cunningham 194 noted that \i N = A m + A, 

 4N + 1 = (ra + n + I) 2 + (m n) 2 ; conversely, if 47V + 1 has no prime 

 factor 4& 1, it is a sum of two squares and hence N is a sum of two A's. 



Crofton 195 noted that 



9 A. + 1 = A(3& + 1), 4 A fc + 4 A; + 1 = (k - Z) 2 + (k + I + I) 2 - 

 Christie employed A m + A m +i = (m + I) 2 to get 



An + A 2 + B* + ' = An + ( A+l + A n+2 ) + + ( A 2 n-2 + A 2 n-l) 



= (An + An+l) + (An+2 + A n+3 ) + ' ' ' + A 2 n-l 



= A 2 n-i + a 2 + /3 2 + 



W. A. Whitworth 196 gave rules, depending on the convergents to the 

 continued fraction for V2, to solve A = D or A = 2 A', equivalent to 

 known rules to solve u 2 2v 2 = d= 1. 



E. Lemoine 197 called a number N decomposed into its maximum tri- 

 angular numbers A, and m the index of N, if N = A\ + + A m , where 

 Ai is the largest A = N, A 2 the largest A = N AI, A 3 the largest 

 A = N AI A 2} etc. If Y m is the least number of index m, 



Y m = JY^Y^ + 3), 2- 1 F m = (Y l + 3)(F 2 + 3) -(Y^ + 3). 



1 9 Mathesis, (2), 8, 1898, 52^. Cf. Tits. 223 



180 Die Arith. der Quadratischen Formen, 1, 1898, 154-162. 



191 Math. Quest, Educ. Times, 68, 1898, 84. 



lUd., 69, 1898, 114. 



Ibid., 70, 1899, 119. 



* Ibid., 71, 1899,33. 



*Ibid., 69. 



186 Ibid., 73, 1900, 32-3. 



187 Assoc. franc, avanc. sc., 1900, II, 72. 



