CHAP. I] POLYGONAL, PYKAMIDAL AND FIGURATE NUMBERS. 11 



bach inferred that every number is of the form 2 A db D, and incorrectly 

 (Euler, p. 134) that every number is a sum of three triangular numbers. 

 Euler, June 30, 1742 (p. 133) noted that every number is of the form 

 y 2 + y - x* = 2 A y - x 2 . April 6, 1748 (pp. 447-9, 468), Goldbach stated 

 that every number can be expressed in each of the eight forms 



+ 20'+ A, D+2D / + 2A, D + D' 



2D + A + 2 A', etc. 



June 25, 1748 (pp. 458-460), Euler gave the identity 



a 2 + a 6 2 + 6 _/ d 2 + d\ 



~2 --- 1 --- 2" I ~2~~J' ^ra=--d + e, b = d - e. 



Hence [Fermat's 36 theorem] every n is a sum of three A's implies 



n = D + 2A + A'. 



Euler expressed his belief that every number of the form 4n + 1 is a sum 

 GO of three squares, whence n = D + D' + 2 A. Replacing n by 2n, we 

 see that every n = D + D' + A. Euler gave fourteen such formulas. 

 June 9, 1750 (p. 521), Euler remarked that an algebraic discussion of the 

 theorem that any number n is a sum of three triangular numbers is of no 

 help, since the theorem is not true if n is fractional (unlike the theorem on 

 3]). Dec. 16, 1752 (p. 597), Euler noted as facts, of which he had no proof, 

 that every prime Sn + 1 or Sn + 3 is of the form x 2 + 2y 2 , whence if 

 n=|=n + A,8n+l+ prime, and if n 4= 2 A + A', Sn + 3 =t= prime. Also 

 (p. 630), if n * D + 2 A, 4n + 1 + prime. 



April 3, 1753 (pp. 608-9), Euler treated the problem [of Fermat 43 ] to 

 find a number z which is polygonal in a given number of ways. Let n be 

 the number of sides of the polygonal number, x its root. Then 



2z = (n- 2)x 2 - (n - 4)z, n = 2 - - + 2( * ~ ^ . 



x x 1 



Thus 2z must be divisible by x, and 2z 2 by x 1. Hence we desire 

 two numbers differing by 2 which have divisors differing by 1. For example, 

 450 and 448 have such divisors 3 and 2, 5 and 4, 9 and 8, 15 and 14. Thus 

 225 is a square, 8-gon, 24-gon, and 76-gon. 



Euler 60 noted that, if 4n + 1 is a sum of two squares, Sn + 2 is a sum 

 of two odd squares (2x + I) 2 , (2y + I) 2 , whence n = A* + A tf . S. Realis 60a 

 noted that conversely this expression for n implies 



+ 1 = (x + y + I) 2 + (x - 2/) 2 . 



In Ch. Ill are cited Euler's 3 theorem H(l - x k ) = E(~ I) y z 7 '> where 

 p = (3j*j)l2 is a pentagonal number, and theorems by Legendre, 23 

 Vahlen, 150 and von Sterneck, 169 on the partitions of N, in which an excep- 

 tional role is played by the N's which are pentagonal or triangular. 



60 Novi Comm. Acad. Petrop., 4, 1752-3 (1749), 3^0, 34; Comm. Arith. Coll., I, 164. 

 600 Nouv. Ann. Math., (3), 4, 1885, 367-8; Oeuvres de Fermat, IV, 218-20. 



