CHAP. I] POLYGONAL, PYRAMIDAL AND FIGURATE NUMBERS. 31 



Several writers 183 found the first six integers n making n(n + l)/2 a 

 square. Several 184 proved that the difference of the roots of two successive 

 triangular numbers, each a square, equals the sum of two successive integers 

 the sum of whose squares is a square. 



A. Boutin 185 reduced x z = A v + 1 to p z 2q z = 1 by setting 

 2x = 3q =F p, y = (k - l)/2, k = 3p =F 2q [Euler 79 ]. 



G. de Rocquigny 186 noted the identities 



(5n 1)8 = !l) + 2?L 



\ A (a 2 + a - 1) + A (a 2 - a - 1) } { A(6 2 + b - 1) + A(6 2 - b - 1) } 



= A(a 2 6 2 + ab - 1) + A(a 2 6 2 - ab - 1), 



(A(7a + l) + A(a- 1)}{A(76 + 1) + A (6 - 1)} 



= A(7c + 1) + A(c - 1), c = 5ab + a + 6, 



and expressed n 6 , n + n z + n 3 + ft 4 , n 3 + ft 4 + ft 5 + ft 6 and n + + n 6 

 as sums of three triangular numbers, etc. 



A. Boutin 187 solved A x -i + A n = y z by setting x = an 6, y = an /3. 

 Then 



a 2 + 1 = 2a 2 , 6 2 + 6 = 2/5 2 , 4a/3 + 1 = a(26 + 1), 



which are solved by means of recursion formulas. 



A. Berger 188 proved many relations and inequalities involving the rth 

 w-gonal number (3) designated by P(m, r). If | x \ < 1, 



+' (a - 3)s 



' 1 - x ' > - r)x - 



He evaluated Sl/P(a, r), where r ranges over all integers for which P(a, r) 

 takes positive values and each but once. If a ^ 3, | x < 1, e = 1, 



oe . +00 



fl (1 - Z<- 2 >0(1 + &W-***) (1 + eo^- 2 "- 1 ) = Z e"x P(a ' r \ 



r=l r= 



combinations of special cases of which give 



na - af) ( - i)r = z x p(6 ' r \ ft(i - xo = z (- iyx f(5 ' r) . 



r= 09 r=l r= oo 



Let ff(K) be the sum of the divisors of k, and \f/(k) the excess of the sum of the 

 odd divisors of k over the sum of the even divisors. Then 





log Z (- i) V^ = - Z 



A:=J C r=0 



3 Amer. Math. Monthly, 3, 1896, 81-2; Math. Quest. Educ. Times, 65, 1896, 53; 69, 1898, 



51. 



184 Amer. Math. Monthly, 4, 1897, 187-9. 

 188 Mathesis, (2), 6, 1896, 28-29. 

 186 Mathesis, (2), 7, 1897, 217-221. 



187 Ibid., 269-270. 



188 Nova Acta Soc. Sc. Upsaliensis, (3), 17, 1898, No. 3. 



