CHAP. Vi: SUM OF TWO SQUAKES. 253 



A. GeVardin 154 discussed the solution of 

 (Hte + m) 2 + (Wy + p) 2 = lOOa, a = b 2 + d 2 , m < 10, p < 10. 



Since m 2 + p 2 = 20/i, we have m = 2, p = 4 or 6; m = 4 or 6, p = 8. 

 These cases are treated in turn. To solve (pp. 89-90) x 2 + y- = a 2 + 6 2 

 set x = a + mh, b = y + h, m(x + a) = b + y> Then 



h = 2(y - am)/(m 2 - 1), 

 and the general solution is said to be 



(am 2 - 2my + a) 2 + y 2 (m 2 - I) 2 = a 2 (m 2 - I) 2 + (ym 2 - 2am + y) 2 . 



W. Sierpinski 155 gave a long proof that, if A(x) is the number of pairs of 

 integers u, v for which u 2 + v 2 ^ x, A(x) = irx + 0(x 113 ), for defined as 

 in Landau, 179 while TT is the usual constant. 



E. Jacobsthal 156 proved that, if p is a prime = 1 (mod 4), p = a 2 + b 2 , 

 where, in terms of Legendre's symbols, 



\ 

 1 , 



J 



where r is any quadratic residue (as 1) of p, and n any non-residue. Also, 

 a = (p 3)/2 (mod 8). Proof is given of formulas, equivalent to Gauss', 44 

 for the residues of a, b modulo p. 



Identities 157 solving a 2 + b 2 = 2c n have been given. 



W. Sierpinski 158 evaluated sums like 



n=l 



where r(n) and rg(w) denote the number of decompositions of n into 2 

 and 8 squares. 



* E. N. Barisien 159 expressed 2 n as a ratio of two 03. 



J. Sommer 160 applied ideals to show that every prime 4n + 1 is a E] . 



L. Aubry 161 cited known results. 



G. Bisconcini 162 proved that n is a El if and only if n contains no odd 

 power of a prime 4fc 1, and deduced all decompositions of p r as a (2, 

 given that of the prime p = 4& + 1. He 163 proved that, if pi is a prime 

 4k + 1, p* 1 p" has 2 m ~ 1 proper decompositions into 120 ; also Gauss' 37 

 theorem. He treated (pp. 68-80) the decomposition of fractions into 

 one of the forms x 2 y 2 . 



154 Sphinx-Oedipe, 1906-7, 112-9. 



155 Prace mat.-fiz., Warsaw, 17, 1906, 77-118 (Polish). See papers 179, 180, 189, 198-203. 

 186 Anwendungen einer Formel aus der Theorie der quadratischen Reste, Diss. Berlin, 1906, 13; 



Jour, fur Math., 132, 1907, 238-245. 

 157 L'intermediaire des math., 13, 1906, 62, 184; 14, 1907, 72. 



168 Prace mat.-fiz., Warsaw, 18, 1907, 1-60 (Polish). Reviewed in Jahrb. Fortschritte Math., 



38, 319-21; Bull, des Sc. Math., (2), 37, II, 1913, 30-31. 



169 Bull. Sc. Math. Elem., 12, 1907, 262-6. 



160 Vorlesungen iiber Zahlentheorie, 1907, 112, 123-4. French transl. (of revised text) by A. 



Levy, 1911, 105, 117-9. 



161 L'enseignement math., 9, 1907, 421. 



162 Periodico di Mat., 22, 1907, 270-285. 



163 Ibid., 23, 1908, 9-23. 



