48 Sum of one Square and two Triangular Numbers. 

 Consider the identity (N=) 



+ K- +£ ?)(-'" + ^ +1 ) 



and observe that, if p be of the form 4?i + 3 or An, \p ,jt? + 1 is 

 even, and that, if jt> be of the form An + 1 or An + 2, it is uneven. 

 Then (N being uneven) if m be uneven, and consequently 

 ip -p + 1 and J<7 . q + 1 be both even or both uneven, then, accord- 

 ing as p and q are both of the same form or of different forms, 

 the first or second transformation gives rise to the sum of an 

 even square, an even triangular number, and an uneven trian- 

 gular number. And, conversely, an even square and two tri- 

 angular numbers, one even and one uneven, are transformed 

 into an uneven square and two triangular numbers, both even 

 or both uneven. This is evident on examining the different 

 cases. We thus have the curious theorem that to every re- 

 presentation of an uneven number as an even square and two 

 triangular numbers, there corresponds a representation as an 

 uneven square and two triangular numbers, and vice versa ; 

 and we further see that the presence of a zero square corre- 

 sponds to a case of equality of the two triangular numbers *. 

 If, therefore, all the representations of an uneven number as 

 the sum of a square and two triangular numbers be written 

 down, in half the number the square is even and in half 

 uneven. Stated analytically, the theorem is 



(1 + 2q* + 2q™ + &C.)(1 + q e + q 10 + &c.)(q + q* + q™ + &c.) 



= (s + f + <f° + &c -)(i + q 6 + q 10 + &C.) 2 



+ (3 + 2* + <f° + J&Xff + 9 d + 9 15 + &c.) 2 , 



* Of course is to be treated as a square, and also as an even triangular 

 number whose root (calling n the root of \ n . w+1) is j and the square 

 numbers that are also triangular are to be treated both as squares and tri- 

 angular numbers. As an example of the theorem, take N=31, and 

 25+6+0, 25+3+3, 9+21+1, 1 + 15+15 transform into 4+21 + 6, 

 0+28+3, 16+ 15+0, 0+21+10 respectively (writing the square first in 

 each partitionment). 



