CHAP. I] POLYGONAL, PYRAMIDAL AND FIGURATE NUMBERS. 15 



summands. Next, bring in the summand C and let c be the least posi- 

 tive integer such that C + c requires t 2 + 1 summands from 1, A, B, C. To 

 produce the numbers ^ D, we need 



t i J-- 



J 2 T 1 



D 



summands, etc. In the case of an infinite series 1, A, B, , the process 

 furnishes a lower limit to the number t of summands. Euler showed that, 

 for series whose nth order of differences are constant, Beguelin's rule is 

 often quite erroneous, but did not treat the series 1, n + d, of polygonal 

 and pyramidal numbers. 



N. Beguelin 74 made a puerile illogical attempt to prove that every num- 

 ber is the sum of three triangular numbers. Admitting the last theorem, 

 Beguelin 75 deduced Bachet's theorem that every integer is a HI . For, 



(5) n = 2(a 2 + a)/2 implies 8n + 3 = Z(2a + I) 2 . 



Adding 1, we conclude that 8n + 4 is a ffl. But it is known that the half 

 or double of a EC is a S3 . Hence 2n + 1 and its product by any power of 

 2 are SI. Since Lagrange had given in 1770 an independent proof of this 

 theorem of Bachet, Beguelin next attempted, but failed completely, to 

 deduce from it the result that every integer is a sum of three triangular 

 numbers A. On p. 338, he gave the equivalent formulas 



ff = ^ + ^-ir^, 4? + 1 = (a - b} 2 + (a + b + I) 2 . 



He concluded without adequate proof that every number is a sum of a A 

 and two squares, and also is A + 2A' + 2 A" (p. 345); further, that 

 every integer = 1, 2, 3, 5 or 6 (mod 8) is a 05, later proved by Legendre 19 

 of Ch. VII. A fitting sample of the lack of insight of Beguelin is furnished 

 by his final theorem* (p. 368) : If any number 4m + 1 is a sum of three 

 squares [each 4= 0], it is composite [but 17 = 9 + 4 + 4 is prime]. 

 Curiously enough, he supposed he had verified the theorem for all numbers 

 < 200; but his tables (pp. 363-4) imply that he assumed that a number 

 can be expressed in a single way as a sum of squares. On this he based a 

 new " proof " that every prime 4m + 1 is a G3. 



L. Euler 76 noted the result (5). 



Euler 77 noted that ^ is not the sum of t three fractional triangular numbers 

 (x 2 + x)/2, since 7 is not the sum of three odd squares (2x + I) 2 - But every 



74 Nouv. M&n. Acad. Berlin, annexe 1773, 1775, 203-215. 



75 Nouv. Mem. Acad. Berlin, annee 1774, 1776, 313-369. 



* One error is that if the sum of three A's, each 4= 0, is of the form 3i> + 2, then v is 

 not divisible by 3, assumed to follow from the converse in 50. But 45 + 10 + 1 = 2 

 (mod 9). 



76 Acta Acad. Petrop., 4, II, 1780 [1775], 38; Comm. Arith. Coll., II, 137. Euler 11 of Ch. VII. 



77 Opusc. Anal., 2, 1785 (1774), 3; Comm. Arith. Coll., II, p. 92. 



