16 HISTORY OF THE THEORY OF NUMBERS. [CHAP. I 



number N is the sum of four fractional pentagonal numbers (3# 2 x)/2, 

 since 



24N + 4 = Za 2 = 2(6z - I) 2 , x = , 



To prove the theorem that any number is the sum of three integral tri- 

 angular numbers A, it would be sufficient to show that the coefficient of 

 every term x k in the expansion of 



(1 + X + X 3 + X 6 + + X* + - - -) 3 



is not zero; similarly for squares, pentagons, etc. [Euler 69 ]. Let the 

 polygonal numbers with ?r sides be 0, a = 1, /3 = TT, 7 = 3?r 3, 

 and denote by [n~] the coefficient of x n in (1 + x a + xP + )*. Euler 

 proved by logarithmic differentiation the recursion formula 



- (n - j) } o - n. 



/=o,/J,... 



F. W. Marpurg 78 treated (pp. 185-257) polygonal numbers, giving special 

 cases of formula (1) of Diophantus, pyramidal numbers and central poly- 

 gonal numbers, viz., unity more than the number of the angles and division 

 points on w-gons drawn about a common mid point. Also (p. 307) poly- 

 hedral numbers, the rth hexahedral, octahedral, dodecahedral and icosa- 

 hedral being 



r 3 , ^(2r 2 + l), ^(9r 2 -9r + 2), r - (5r 2 - 5r + 2) . 

 Euler 79 proved that (x 2 + x)/2 is a square ?/ 2 only when 



For n = 0, 1, 2, we get x = 0, 1, 8; y = 0, 1, 6. We have the recursion 

 formulas 



z = 6a: n _i - x n -2 + 2, y n = 6y_i - 2/_ 2 . 



Certain squares x z which exceed (?/ 2 + j/)/2 by unity are given by 

 _ (2A/2 + l)g+ (2V2- l)/3 _ (2V 2 "+ l)g - (2V2 - 



_ 

 4V2 ' . " 4 2 ' 



For n = Q,x = l,y = Q; forn = 1, a; = 4, y = 5. The recursion formula 

 is 



i - rc n _ 2 , n = 6_i - ? n -2 + 2. 



A second series of solutions is obtained by use of these formulas for negative 

 n's. Thus x-i = 2, T/_! = - 3; a;_ 2 = 11, 2/_ 2 = 16. Since the tri- 

 angular number A- m equals A TO _i, we replace y = mbym 1 and get 

 the sets of positive solutions 2, 2; 11, 15; etc. 



78 Anfangsgriinde des Progrcssional Calculs, Berlin, 1774, Book 2. 



79 M6m. Acad. St. P6tersbourg, 4, 1811 [1778], 3; Comm. Arith. CoU., II, 267-9. 



