CHAP. I] POLYGONAL, PYRAMIDAL AND FIGURATE NUMBERS. 29 



[But the denominator 2 on the left should be suppressed. Legendre 94 had 

 already proved a more general theorem.] 



E. Lucas 171 collected results, mostly algebraic, on triangular and figurate 

 numbers. 



E. Catalan 172 stated that every A, not pentagonal, is a sum of fewer 

 than 7 pentagonal numbers. [Catalan. 169 ]] 



T. Pepin 173 gave a proof of Cauchy's formulation of Format's theorem 

 that every integer A is a sum of m + 2 polygonal numbers |m(x 2 x) + x 

 of order m + 2 of which m 2 are or 1. We are to prove that 



A = \m(a 6) + b + r, 



where a = a 2 + + 8 2 , & = + + 5, O^r^m 2, whence 

 a = b (mod 2). Since r can take the values and 1, we may take 6 odd, 

 whence 



4a - b 2 = 81 + 3 = x* + y* + z 2 , 



x > y > z > 0. Determine integers a, , 5 so that 



a + 7-0-3 = 2/, a + /3 + 7 + 5 = &. 



Then a = S 2 is satisfied. The condition & 2 < 4a is satisfied if 5 > 110, 

 where A = mB + c, < c = m. Hence the theorem is true for all numbers 

 A > 110m. It was verified separately for all numbers = 120m + 16. 



G. Musso 174 proved, by use of geometrical representations, Bachet's 32 

 second formula I, 10, and the generalizations 



p; - s -pi + (s- )pr' + s -pr 2 - (s odd) ' 



.. _ 2 n _ 2 s - 2 



also 



P l = n 2 _ ( n _ J)2 _f_ ( n _ 2)2 _ . . . 1 ? 



E. Catalan 175 gave a shorter proof of Bachet's same formula. 

 G. de Rocquigny 176 noted that, ifa + &-fc = o: + /? + 7 = 



P = ( A a + A & + A c )(A a + A^ + A T ), (A + An+i)(A p + A P+2 ), 



+ 1 6n 4 



are sums of three A's, while n 2 + (2n I) 2 + (2n + I) 2 is a sum of two. 



171 Th6orie des nombres, 1891, 52-62, 83. 



172 Jour, de math, spec., 1892, No. 353. 



173 Atti Accad. Pont. Nuovi Lincei, 46, 1892-3, 119-131. 



174 Giornale di Mat., 31, 1893, 173-8. His P q n is our p n . 



175 Ibid., p. 227. 



176 Mathesis, (2), 4, 1894, 123, 171, 211; (2), 5, 1895, 23, 150, 211-2. Cf. Curjel. 180 



