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



The mean value of V(n) is three times that of ^(n). He found that the 

 probability that two triangular numbers taken at random shall be relatively 

 prime is 



Cesaro 161 stated and E. Fauquembergue 161 proved that 5 and 17 are the 

 only integers whose cubes diminished by 13 are quadruples of triangular 

 numbers. 



G. de Rocquigny 162 noted that, if k = (a 2 + b 2 + a + 6)/2, 



A* = A-i + A + A 6 , (a 2 + I) 2 = 1+ A(a 2 + a) + A(a 2 - a). 



S. Realis 163 used the known fact that, if p is a product of primes 8q + 1, 

 2x 2 + y z = p has integral solutions. Thus 



3(8? + 1) = 2(2a + I) 2 + (26 + I) 2 , 

 so that 3q = 2 A + A'. 



Realis 164 gave various sums like 



Ai + A 3 + A 5 + + A 2n -i = \n(n + l)(4n - 1), 

 A 2 + A 4 + A 6 + + A 2n = \n(n + l)(4w + 5), 

 A 3 + A 6 + A 9 + + A 3 n = f n(n + I) 2 . 



E. Cesaro 165 noted that (n 5 l)/4 = A P + A 9 , n =1= 5, implies that 

 2p -\- 1 or 2q + 1 is composite. 



S. Tebay and others 166 found that the least heptagonal number 

 !(5z 2 ~ 3z) which when increased by a 2 is equal to a square is given by 

 x = 24(19a - 9). 



C. A. Laisant 167 wrote a a for the ath (a + 2)-gonal number pl +2 and 

 gave 



(a + &) = a a + b a + aab, (a + - + l) a = 2a a + aZab. 



E. Cesaro 168 noted that the number of A's prime to n and < 2n(n + 1) 

 is k = nll(l 2 /p) or 2k according as n is even or odd, where p ranges 

 over the odd prime factors of n. 



E. Catalan 169 proved that every A > 1 is a sum of six pentagonal num- 

 bers. For, 170 6(2n + I) 2 = (Gx T I) 2 + (Qy =F I) 2 + 4(62 =F I) 2 , whence 



n(n + 1) 3x 2= Fa; 3?/ 2 =F y /3g 2: Fg\ 

 _ 2 ~ 2 + 2 + V 2 /' _ 



161 Mathesis, 6, 1886, 23; 7, 1887, 257-9. 



162 Ibid., 6, 1886, 224. 



163 Nouv. Ann. Math., (3), 5, 1886, 113. 



164 Jour, de math. sp6c., 1888, 94. 

 166 Mathesis, 8, 1888, 75. 



166 Math. Quest. Educ. Times, 50, 1889, 84-5. 



167 Bull. Soc. Philomathique de Paris, (8), 3, 1890-1, 29-30. 

 188 Mathesis, (2), 1, 1891, 95-96. 



169 Assoc. frang. av. sc., 1891, II, 201-2. 



170 Recherches sur quelques prod, indcf., Mdm. Acad. Roy. Belgique, 40, 1873, 61-191, formula 



393. 



