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



For analogous theorems on sums of consecutive squares or the sum of 

 the squares of the first n odd numbers see papers 70, 76, 81, 86, 87, 100, 

 and 103 of Ch. IX, and Brocard 92 of Ch. XXIII. 



W. Goring 139 proved by use of infinite series that 2 A + 6 A 7 + 1 can 

 always be represented by the form a 2 -f 36 2 . 



J. W. L. Glaisher 140 noted that every representation of an odd number as 

 a sum of an even square and two triangular numbers corresponds to a 

 representation in which the square is odd, since 



for p, q both even or both odd, with a similar identity if one is even and 

 the other odd. 



Glaisher 141 stated that every triangular number is a sum of three 

 pentagonal numbers. 



D. Marchand 142 noted the relations 



Pi = Pl~ l + r 2 , Pi = 2?;-' + r 2 , Pi + Pi + + Pi = rpl. 

 Marchand 143 gave identities like 



A (fy + 1) = Afo) + (2y + I) 2 , 

 (* + I) 5 - x 5 = A(y) + A(3y + 1) = 2A(y) + (2y + I) 2 , 



where y = x 2 -f- x, and (p. 105) discussed triangular numbers which are 

 squares. 



E. Lucas 144 asked when (A* + + A*)/(Ai + + A n ) is a square. 

 S. Re"alis, E. Catalan and others 145 investigated numbers simultaneously 



squares and triangular. S. R6alis stated and E. Cesaro 146 proved that the 

 square of every odd multiple of 3 is a difference of two A's prime to 3, 

 9(2n + I) 2 = A (9/i + 4) - A(3n + 1). D. Marchand 147 gave the gen- 

 eralization that the square of any odd number is the difference of two 

 relatively prime triangular numbers (with sides 3x + 1 and x). C. Henry 148 

 proved a like result for the product of any odd square by any number. 



S. Re'alis 149 stated that the theorem that every integer n is a sum of 

 three A's implies that n is a sum of four A's of which two are consecutive 

 and that n is a sum of four A's two of which are equal. 



Math. Annalen, 7, 1874, 386. 



o Phil. Mag., London, (5), 1, 1876, 48. 



141 Messenger Math., 5, 1876, 164-5. 



"* Les Mondes, 42, 1877, 164-170. 



lt * La Science des nombres, 1877. 



144 Nouv. Corresp. Math., 3, 1877, 433. 



/6id., 4, 1878, 167; 5, 1879, 285-7; Math. Quest. Educ. Times, 30, 1879, 37. 



146 Nouv. Corresp. Math., 4, 1878, 156. 



147 Nouv. Ann. Math., (2), 17, 1878, 463. 

 " Ibid., (2), 19, 1880, 517. 



d., (2), 17, 1878,381. 



