CHAP. XI] LIOUVILLE'S SERIES OF EIGHTEEN ARTICLES. 331 



and u, v any even integers. For f(x) = x*, we deduce from (D) that 



which gives the number of decompositions of 4m into s + 2V, where s 

 and <T are sums of 4 odd squares. 



For m any integer > 1, let m = m' + m". Liouville stated the follow- 

 ing case of ( f ) : 



Z ( d Z lf(d' - d"} - f(d' + d")]} = /(O) { fi(m) - r (m) } 



- - S/(4){2f() + <* - 25-1} - 2Z'{/(2)+/(3)+. . -+/(d-l)}, 



where f (m) is the number of factors of m and the accent on the final sum- 

 mation sign signifies that a term f(k) is to be suppressed when k is a divisor 

 of d. For f(x) = x z and m a prime, (H) gives 



(HO ZfiCm'KiCro") = A(m 2 - l)(5m - 6). 



This result may be used to prove the theorem of Bouniakowsky that 

 any prime m of the form 16/b + 7 can be decomposed into 2x 2 -f- p 4l+l y z 

 in an odd number of ways, where p is a prime 4X+1 not dividing y. 



Liouville 3 stated that, if f(x, y) is unaltered by the change of the sign 

 of x or y, 



2 Z { d Z, C/(d' - 2 a "cT, 5' + 5") - /(d' + 2*"d", d f - ")]} 

 (d) = S {/(d, 0) + 2f(d, 2) + 2/(d, 4) +-+ 2/(d, 6 - 1) - 4f (d, 0) } , 



d 



where the first summation extends over all decompositions m' + 2 a "m" of 

 m. If f(x, y) reduces to a f unction /(x) of x only, (d) becomes (F). If it 

 reduces tof(y), (d) becomes (D). To pass from (D) to (E), take 



f(x) = F(x + 1) - F(x - 1). 



In (d) take/ to be ( l) 2//2 /(x), where /(x) is an even function. Then 



' - 2-V) +f(d f + 2- 



For 2*w = 2 a 'w / + 2 a "m", 

 S{2[/(2 a 'd' - 2 a V, 5' + d") -f(2*'d r + 2 a "d", 5' - 5")]} 

 = Zd{/(0, 2d) + 2/(0, 4d) + 4/(0, Sd)+ "-+ 2 Q - 1 /(0, 



+ 2 {/(2-d, 0) + 2/(2d, 2) + 2/(2V, 4) + 



+ 2/(2-d, 5 - 1)} - 2 a 2df(2*d, 0), 



which reduces to (G) for/(z, ?/) = /(x). Formula (H) is a special case of 

 Z ( Z C/(d' - d", d' + 6") - f(d' + d", 5' - 6")]} 



t'+m"=n 



(f) = (<* _ i){/( , d) -/(d, 0)} + 2E'{/(, 2) + ... 



+ /(,d-l)} -2Z'(/(2, 5)+ ... +/(d-l, 6)}, 

 where the accent indicates that f(8, y} is to be suppressed if y is a divisor 



3 Jour, de Math., (2), 3, 1858, 273-288. Fifth article. 



