CHAP. VI] SUM OF Two SQUARES. 257 



M. Weill 196 noted that the product of p sums of two squares is a sum of 

 two squares in 2 p ~ l distinct ways. 



M. Chalaux 197 proved Girard's theorem by induction using the fact that 

 if a prime is a C3 and divides a sum of two relatively prime squares, the 

 quotient is a sum of two relatively prime squares. 



E. Landau 198 proved his 179 former theorem by means of a new simplifi- 

 cation of Pfeiffer's method (cf. pp. 305, 322 of Vol. 1 of this History). 

 He 199 next considered the lower limit a of the constants for which A (x) = 

 TTX + 0(x a ), and proved that a: = J. Later he 200 proved a theorem on the 

 number of lattice points in certain regions which is a generalization of the 

 main theorem applied in his 179 above papers. 



* K. Szilysen 201 stated empirically an asymptotic formula for the num- 

 ber of pairs of integers for which x* + y* ^ N, a formula already proved 

 by Lipschitz. 



M. Rignaux 202 announced a table in manuscript of the decompositions 

 as a [Zl of the 3908 decomposable numbers < 10000. 



G. H. Hardy 203 deduced Landau's 179 theorem very simply by two dif- 

 ferent methods from the theorems in Hardy's 190 former paper. If 204 ai, 



a m are primes of the form 4& -f 1, there are 4(n + l) w sets of solutions of 

 z 2 + y 2 = (i oz - - a m ) n , in 2 m+2 of which x and y are relatively prime. 



On the number of solutions of x 2 + (4?/) 2 = n, see Nasimoff 68 of Ch. 

 XIII. On x 2 + 2/ 2 = (m 2 + n 2 ) z 2 , see papers 142-5 of Ch. XIII and the 

 cross-references given there. On 1 + x 2 = 2y*, see Euler 7 of Ch. XIV and 

 Cunningham 79 of Ch. XX. In Ch. XVII are given reports on papers on 

 a number and its square both sums of two consecutive squares ; cf . Meyl 30 

 of Ch. IV. On x 2 + n 2 * y 3 , see Pepin 10 and Hayashi 61 of Ch. XX. On 

 x + y = D, z 2 + y* = z 4 , see papers 40, 48, 50, 52, 54-56, 63 of Ch. XXII. 

 On systems of equations including x z + y z = z 3 , see papers 353, 363, 368 of 

 Ch. XXI. Equal sums of two squares occur on p. 37, p. 206; in paper 

 107a of Ch. VII; 18 of Ch. XIII; papers 21, 35, 45, 62, 80 of Ch. XV; 7, 

 9, 18, 20 of Ch. XVIII; 4, 13, 15, 33, 37, 42, 46-50, 75, 102, 133, 149 of Ch. 

 XIX; 177ofCh.XXII; 45of Ch.XXIV. In Vol. I were cited the papers 

 by Euler 3 - 7 and Gauss, 13 pp. 381-2, containing tables of primes and factors 

 of numbers x 2 +y 2 ; by Lucas 53 and Catalan, 61 pp. 402-3, on special num- 

 bers which are 12; by Liouville 28 , p. 286; and various papers, pp. 360-1, 

 on factoring numbers which are El in two ways. 



196 Nouv. Ann. Math., (4), 16, 1916, 311-4. 



197 Ibid., (4), 17, 1917, 305-8. 

 198 G6ttingen Nachrichten, 1915, 148-160. 

 199 Ibid., 161-171. 



m lbid., 209-244; 1917, 96-101. Cf. Revue semestrielle, 27, I, 1918, 16, 18. 

 201 Math, es terms. ertesito (Hungarian Acad. Sc.), 35, 1917, 54-6. 

 202 L'interme"diare des math., 25, 1918, 143; 26, 1919, 54-55. 

 "O'Proc. London Math. Soc., (2), 18, 1919, 201-4. 

 r. Math. Monthly, 26, 1919, 367-8. 



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