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



J. Plana 123 wrote 4 for the left member of (6). By expanding the second 

 member as a power series in q and examining the earlier terms, he verified 

 that 



oo 



n=l 



where <r(k] is the sum of the divisors of k. Hence any integer n is a sum of 

 4 triangular numbers in a(2n + 1) ways. Give to 3 the notation of a 

 power series in q, multiply it by and compare with the above series for 4 ; 

 we get a recursion formula for the coefficients of 3 . He states without 

 proof that the coefficient of every power of q is not zero, and so concludes 

 that every integer is a sum of three triangular numbers. 



F. Pollock 124 verified for small values that any number may be expressed 

 in the form s s', where s and s' are sums of two triangular numbers. 

 Now s is always the sum of a square and the double of a triangular number. 

 Thus the theorem is that 



(7) a 2 + a + b 2 - (m 2 + m + w 2 ) 



represents any number. Take p 2 c 2 c + q as the number. Then 



a 2 + a + 6 2 + c 2 + c = m 2 + m + n 2 + p 2 + q. 

 Double and add unity. Thus A = M + 2q, where 

 A = 2a 2 + 2a + 1 + 26 2 + 2c 2 + 2c, M = 2m 2 + 2m + 1 + 2n 2 + 2p 2 . 



Since 5 is arbitrary, it is concluded that any odd number can be represented 

 by either of the forms A or M . But M is the sum of four squares. 

 Again, represent p 2 \ (c 2 + c) + q by (7). As before, 



2a 2 + 2a + 1 + 26 2 + c 2 + c 



represents any odd number 2n + 1. But a 2 + a + & 2 is the sum of two 

 triangular numbers. Hence n is the sum of three triangular numbers. 



Pollock 53 of Ch. VIII noted that the theorem that every number 4n + 2 

 is a sum of four squares implies that every integer n is a sum of four A's. 



J. Liouville 125 considered the partition of any number into a sum of 

 ten triangular numbers. 



S. Bills 126 solved A* + A y = A a by setting y = a xr/s and finding 

 x rationally. 



E. Lionnet stated and V. A. Lebesgue and S. Re*alis 127 proved that 

 every integer is a sum of a square and two A's, also a sum of two squares 

 and a A. 



A. Hochheim 128 gave linear relations between polygonal and polyhedral 

 numbers. 



128 M6m. Acad. Turin, (2), 20, 1863, 147. 



124 Proc. Roy. Soc. London, 13, 1864, 542-5. 



125 Comptes Rendus Paris, G2, 1866, 771. 



126 Math. Quest. Educ. Times, 6, 1866, 18. 



127 Nouv. Ann. Math., (2), 11, 1872, 95-6, 516-9; (2), 12, 1873, 217. 



128 Archiv Math. Phys., 55, 1873, 189-192. 



