246 HISTORY OF THE THEORY OF NUMBERS. [CHAP. VI 



If the path is symmetrical, there are further conditions. He gave a history 

 of the subject. 



N. V. Bougaief 109 applied elliptic functions to the decomposition of 

 numbers into squares (with relation to Jacobi's 47 Fundamenta Nova). 



E. Fauquembergue 110 noted that a cube =|= 1 is never a sum of squares 

 of two consecutive integers. 



E. Cesaro 111 considered the function $(ri) = 2/(a), where a ranges over 

 all the positive integers for which n a 2 is a square. Then 



= t, 



j-l .7=1 



For /(a;) = 1, \f/(n) is the number of positive integral solutions of x 2 + y 2 = n\ 

 then 2^(j) equals n?r/4 asymptotically, whence the number of ways of 

 decomposing a number into a sum of two squares is in mean 7r/4. 



T. J. Stieltjes 112 states that if f(ri) is the number of solutions of 

 s 2 + y 2 = n, and if ju is the largest odd integer ^ Vn, then 



1 1 0* ~ I)* I/ JyiN 



+ 4^ 2 - , n-l(mod4), 



4(2* +1) 



-- 

 2 _. , Nf rn 



= 8 S ( ~ 1} L 8(2^ + 1 



/(n) 



5) 



8(2^+1) mn Y 



where, in the last, k = [_^(^n + 4 1)]. If ^(x) is the sum of the odd 

 divisors of x, 



0(1) + 0(5) + - + 0(4n + 1), 0(1) + 0(3) + + 0(2n - 1), 



0(1) + 0(2) + + *(n) 



are expressed as sums of greatest integers. 



T. Pepin 113 proved that, if m is an odd number not a square, 



m<r(m) = 2^ (2 + (- l) w - n }(5w 2 - 



where X(k) is the sum of the odd divisors of k and <r(k) is the sum of all 

 the divisors of k. Let m be a prime 4Z + 1. Hence 



1 = 2(20 M 2 - - m)ff(m - 4 M 2 ) (mod 2). 



Thus among the differences m 4ju 2 occur an odd number of squares, so 

 that m is a E]. 



109 Math. Soc. Moscow, 11, 1883, 200-312, 415-456, 515-602; 12, 1885, 1-21. 



110 Nouv. Ann. Math., (3), 2, 1883, 430. 



111 M6m. Soc. Roy. Sc. de Lifegte, (2), 10, 1883, No. 6, pp. 192-4, 224. 



112 Comptes Rendus Paris, 97, 1883, 889-891. 



113 Atti Accad. Pont. Nuovi Lincei, 37, 1883-1, 41. 



