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



is on grounds not proved. As to the revised version by Gough 87 , Bar- 

 low noted (p. 44) that the argument is correct and trivial to within 12 

 lines of the end; the proof is valid for numbers ^ 3w, but not for those 

 > 3m. 



E. Barruel 88 noted that sums of 1, 2, 3, give the triangular numbers 

 1, 3, 6, , whose sums give the pyramidal numbers 1, 4, 10, 20, , etc. 

 Forming these sums, we get the general triangular and pyramidal numbers 

 n(n + l)/2, n(n + l)(n + 2)/6, etc. Application is made to prove the 

 ordinary rule for deriving a binomial coefficient from the preceding coeffi- 

 cients. 



F. T. Poselger 89 gave (pp. 19-31) various properties of numbers from 

 the writings of Theon of Smyrna, and (pp. 32-60) gave algebraic expres- 

 sions for polygonal and figurate numbers, with a discussion of arithmetical 

 series of general order. 



P. Barlow 90 noted that, if N is a sum of five pentagons (3w 2 u)/2, 

 and M a sum of six hexagons 2z 2 x, then 



+ 5 = E (6w< - I) 2 , SM + 6 = g (4z, - I) 2 . 



In general, if P is a polygonal number of a + 2 sides, Fermat's 36 theorem is 

 equivalent to 



a+2 



SaP + (a + 2) (a - 2) 2 = g (2** - a + 2) 2 . 



He erred 100 - 107 (p. 258) in saying that no triangular number > 1 is pentagonal. 



J. Struve 91 discussed figurate numbers (binomial coefficients). 



J. D. Gergonne 92 noted that the number of terms of a polynomial of 

 degree m in n unknowns is (m + ri) I -r- (ml nl). If the latter be designated 

 (m, ri), then (m, ri) = (m 1, ri) + (m, n 1). 



A. Cauchy 93 gave the first proof of Fermat's theorem that every number 

 is a sum of m w-gonal numbers. The proof shows that all but four of the 

 ra-gons may be taken to be or 1. The auxiliary theorems on sums of 

 four squares will be quoted in Ch. VIII. In the simplified proof by 

 Legendre, 94 the case m = 3 is not presupposed, as was done by Cauchy. 

 Moreover, Legendre proved (p. 22) in effect that every integer > 28(m 2) 3 

 is a sum of four w-gonal numbers if m is odd; while, for m even, every 

 integer > 7(m 2) 3 is a sum of five m-gonal numbers one of which is or 1. 



87 New Series of the Math. Repository (ed., T. Leybourn), 3, 1814, II, 1-7. 



88 Correspondance sur 1'Ecole Imp. Polytechnique, Paris, 2, 1809-13, 220-7. 



89 Diophantus iiber die Polygonzahlen uebersetzt, mit Zusatzen, Leipzig, 1810. 



90 Theory of Numbers, 1811, 219. Minor applications in papers 17-19 of Ch. IX. 



91 Uber die gewohnlichcn fig. Zahlen, Progr. Altona, 1812. 



92 Annales de Math, (ed., Gergonne), 4, 1813-4, 115-122. 



93 Me"m. Sc. Math, et Phys. de 1'Institut de France, (1), 14, 1813-15, 177-220; same in 



Exercices de Math., Paris, 1, 1826, 265-296. Reprinted in Oeuvres de Cauchy, (2), VI, 

 320-353. J. des Mines, 38, 1815, 395. Report by Cauchy, Bull. Sc. par Soc. Philo- 

 matique de Paris, (3), 2, 1815, 196-7. 



94 The'orie des nombres, 1st supplement, 1816, to the 2d edition, 1808, 13-27; 3d ed., 1830, 



I, 218; II, 340; German transl. by Maser, II, 332. 



