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



= e ^ d 4, t<d 1. After this expansion of the argument by 

 Beguelin, we are ready to admit that if e is in one of the intervals 1 to A, 

 AtoB,B to C, it is a sum of d + 2 or fewer terms 1, A, B. He treated four 

 more intervals with a rapidly increasing number of "doubtful cases" for 

 which linear relations between the polygonal numbers were employed, 

 and found in every case that t ^= d + 2. But he finally admitted (p. 405) 

 that this method does not lead to a proof of the general theorem of Fermat. 

 On p. 411, Beguelin stated without proof the erroneous generalization 

 []cf. J. A. Euler, 68 L. Euler 73 ] that any number is the sum of at most 

 t = d + 2n 2 terms of the series 



(n + l)(n + 2)(n + 3d) 



a series whose nth order of differences are constant and equal to d. For 

 n = 2, we have the case of polygonal numbers just considered. For n = 3, 

 we have the pyramidal numbers P r d+2 for r = 1, 2, 3, ; forn = 4, their 

 sums, etc. For n = 4, d = 1, the series is 1, 5, 15, 35, 70, and the 

 theorem gives t = 7, whereas 8 terms are evidently required to produce the 

 sum 64 (since 4 terms must be unity), as expressly mentioned on p. 412. 

 Thus Beguelin contradicts himself in his generalization of Fermat's theorem 

 to pyramidal and figurate numbers. 



L. Euler 73 probably overlooked the last remark, since he stated that the 

 unproved generalization merits great attention. He extended Beguelin's 

 tentative process to any series 1, A, B, . We must employ A + n 2 

 summands 1, A to produce nA 1. Thus if 



nA - 1 ^ B < (n + 1)A - 1, 



we need A + n 2 summands 1, A to produce all numbers 1, 2, -, B. 

 Then 



_ 



Denote by {x} the least integer > x, and by t\ the number of terms 1, A 

 needed to produce 1, , B. Hence 



Bringing in also the summand B, let b be the least positive integer such 

 that B + b requires ti + 1 summands 1, A, B. If C < B + b, we need 

 only ti summands to produce the numbers ^ C. But if C ^ B + b, let 



(m + l)B + b ^ C> mB + b. 

 To produce the numbers ^ C from 1, A, B, we need 



\C - B -b\ 

 ti + m = ti + \- - \= U 



r> } 



"Opusc. Anal., 1, 1783 (1773), 296; Comm. Arith. Coll., II, 27. 



