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



The sum of 2n + 1 consecutive A's equals the product of the middle one 

 by 2n + 1, increased by I 2 + + n 2 . He asked when 



Ai + + An = A. 



E. Barbette 177 noted that the sum T k of the kth powers of the first n 

 triangular numbers equals S k (S + l) k /2 k symbolically, where, after expan- 

 sion, S* is to be replaced by S t , the sum of the tih powers of 1, -, n. 

 The values of TI, T 2 , T 3 are given as functions of n and as functions of the 

 S's. It is shown that T x k = T y r implies x = y, k = r. 



A. Boutin 178 noted that Ax = pA y has an infinitude of solutions if p 

 is not a square. Let 2x = k 1, 2y = z 1. Then k 2 pz 2 = 1 p. 

 Let k = a + p0, z = + a. Then a 2 p0 2 = 1, having an infinitude of 

 solutions if p is not a square. If p = m 2 , the problem has only a finite 

 number of solutions if any. It is impossible if m = 3, 4, 5, 7, 8, 9, 11, 

 12, 13, 15, 16, 17. If m = 4X + 2, x = 4X 2 -f- 4X, y = X is a solution. 



Several 179 solved A* + A v = 2 A z , i. e., 



(2x + I) 2 + (2y + I) 2 = 2(2z + I) 2 . 

 See Ch. XIV. 



H. W. Curjel, 180 to prove de Rocquigny's 176 first statement, took 



a = y z, b = z x, c = x y, a = 77 f, = , 7 = 1, 

 X = x + ZTJ 4- y, Y = z + yy + x, Z = y% + xrj + z, and got 

 P = A(Y - Z} + A(Z - X) + A(Z - F). 



E. Maillet 181 proved the following generalization of Format's 36 theorem 

 on polygonal numbers: If a and are relatively prime odd numbers, 

 a. > 0, every integer A exceeding a certain limit (function of <x, 0) is a sum 

 of four numbers of the form (ax 2 + 0x)/2. We can assign an inferior limit 

 to A such that this decomposition can be made in any assigned number of 

 ways. A like theorem holds if a/2 is an odd integer and one of A, 0/2 is 

 odd and the other even, provided a/2 and 0/2 are relatively prime; also if 

 a/2 is even and 0/2 and A both odd. He proved three complicated theorems 

 stating that every number with certain residues modulo 6 is a sum of at 

 most 5 < 59 (or 5 < 53) numbers of the form (ax 4 + 0z 2 )/2. Later 

 Maillet 182 proved that if tj>(x) = a x 5 + + o s , in which the a's are given 

 rational numbers, is integral and positive for every integer x ^ M, every 

 integer exceeding a fixed limit, depending on the a's, is the sum of at most 

 v positive numbers <j>(x) and a limited number of units, where v = 6, 12, 

 96, or 192, according as the degree of is 2, 3, 4 or 5. Every integer 

 ^ 19272 is a sum (p. 372) of at most 12 pyramidal numbers (x 3 x} 



177 Mathesis, (2), 5, 1895, 111-2. 



178 Jour, de math. 616m., (4), 4, 1895, 179-180. 



179 Math. Quest. Educ. Times, 63, 1895, 40. 



" Ibid., 33-4. Other proofs, (2), 20, 1911, 78-9. 



181 Bull. Soc. Math, de France, 23, 1895, 40-49. 



182 Jour, de Math., (5), 2, 1896, 363-380. 



