CHAP. VII] SUM OF THREE SQUARES. 273 



We have p = a 2 + b 2 , where a and 6 are prime to 3, so that we can set 

 a + & = (mod 3), 



3 



where the three squares are distinct if p > 17. 



L. Aubry 102 proved that not all decompositions of the square of a EC 

 into a O are given by (4). Expressions for x z + ?/ 2 or x 2 + 2?/ 2 as a SI are 

 given on p. 124 and 19, 1912, 11, 188-190. 



H. C. Pocklington 103 noted that, if N = 4w + 1 or 4m + 2, there are 

 properly primitive forms of determinant N that have the quadratic 

 character 1 ; while if N = 8m + 3 there are improperly primitive forms 

 of determinant N which have the character 2. Transform such a 

 form into (6, /, c), where 6 is prime to N. Solve bg 2 = 1 (mod N) for g 

 and let bg 2 + 1 = aN. Then 



N = (a, 6, c, /, g, 0)(6c - / 2 , /<7, - bg) 



is a representation of N by a definite ternary quadratic form of determinant 

 unity. Reducing it in the ordinary way, we get JV = G2 . 



R. F. Davis 104 noted that, if p + q + r = 1, 1/p + 1/q + 1/r = 0, then 



a 2 + 6 2 + c 2 = (pa + #6 + re) 2 + (go + rb + pc) 2 + (ra + pb + gc) 2 . 



E. Landau 105 proved that the number of sets of integers u, v, w for 

 which u 2 + v 2 + w 2 ^ x is ^irx 3 ' 2 + 0(z 3/4+ ), for e > 0. Application is 

 made to the number of classes of positive forms of given discriminant. 



L. Aubry 106 proved that pA 2 = B 2 + C 2 + D 2 implies that p is a sum of 

 three squares; similarly for four squares. 



E. N. Barisien 107 noted various special cases of (3). 



*G. Miihle 1070 solved x 2 -\-y 2 -\-z 2 = g 2 , where g is given; also, x 2 -\-y 2 = g 2 



G. Humbert, 108 by use of an identity involving theta-f unctions, proved 

 that if f(x) is any even function of x, 



where t ranges over the integers occurring in the decomposition of a given 

 number SM + 3 into t 2 + t\ 4- 1\, each t an odd integer > 0, while in the 

 second member the summation extends over the decompositions 



SM + 3 = 4h 2 + ddi (^ > d > 0). 

 The case / = 1 is due to Hermite. 69 He gave a similar result and 



4JV + 3 = t 2 + tl + t\ + 4^ 2 + 411 = 4fe 2 + ddi (t, t lt t 2 odd). 



102 L'interm&liaire des math., 18, 1911, 236. Cf. M. Rignaux, 24, 1917, 35-6. 



103 Proc. Cambr. Phil. Soc., 16, 1911, 19. 



104 Math. Quest. Educ. Times, (2), 21, 1912, 23. 



105 Gottingen Nachr., 1912, 693, 764-9. Cf. Sierpinski. 95 



106 Sphinx-Oedipe, 7, 1912, 81. 



107 Ibid., 8, 1913, 142, 175. 



1070 Ein Beitrag zur Lehre von den pythagoreischen Zahlen, Progr., Wollstein, 1913. 



108 Comptes Rendus Paris, 158, 1914, 220-6; errata, 380. Cf. 157, 1913, 1361-2. 

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