272 HISTORY OF THE THEORY OF NUMBERS. [CHAP, vn 



the number of integers ^ x of one of the forms/ is QcJ C(x}. Since there 

 are [_(x + l)/8] integers 86 + 7 =i x, 



C(x) 5 



J 



lim - . 

 : 6 



2=00 / 



A. Ge*rardin 94 noted that 



(mx ny} 2 + (nx + 2my) 2 = (mx + ny) 2 + (nx)* + (2my) 2 , 

 (x - I) 2 + x 2 + (x + I) 2 = 1 + t 2 , if t 2 = 3z 2 + 1, 



as for (x, = (0, 1), (1, 2), (4, 7), (15, 26), (56, 97), -. To Lucas is 

 attributed 



(12m 2) 2 + 1 = (8m 2) 2 + (8m db I) 2 + (4m) 2 . 



W. Sierpinski 95 noted that if /b is a SI in T 3 (&) ways, 



where < Z 2 -f m 2 + n z ^= x. The number of sets of integers I, m, n 

 satisfying that inequality is ^Trx 3 / 2 + 0(x 5 / 6 ), for as in Landau 179 of Ch. VI. 



E. Landau 96 proved that every positive integer not of the form 

 4 a (8m + 7) is a ED, using the equivalence of every positive ternary quadratic 

 form of discriminant unity to x 2 + y z + z 2 . 



K. J. Sanjana 97 found solutions of the system of equations 



x 2 = y 2 + z 2 + u\ x + y + z + u = 100. 



Let x = a + b, y = a - b. Then z 2 + u z = 4a6, 2a = 100 - z - u. Hence 



(z + 6) 2 + (u + fe) 2 = 26 2 + 2006. 



He took u + 6 = z 6, whence z 2 = 1006. Taking 6 = 1, 4, 9, , he 

 found the solutions 42, 40, 10, 8 and 38, 30, 20, 12. The solution 39, 34, 

 14, 13 was noted by N. B. Pendse. 



H. B. Mathieu 98 stated that the general solution of S = G2 is 



lArB pD, pA + qB=F ID, rA =F IB - qD. 



Welsch" gave I db mv, n =F pv, Im np =F v as the general solution. 

 A. Ge'rardin 100 gave the identity 



(7a 2 + 76 2 - 12a6) 2 = (6a 2 + 66 2 - 14a6) 2 + (3a 2 - 36 2 ) 2 + (2a 2 - 26 2 ) 2 . 



L. Aubry 101 noted the existence of an infinitude of primes each a sum of 

 three distinct squares. Every prime p = 12n + 5>17 gives a solution. 



94 Assoc. frang., 38, 1909, 143-5. 



98 Spraw. Towarz. Nauk (Proc. Sc. Soc. Warsaw), 2, 1909, 117-9. 



98 Handbuch . . . Verteilung der Primzahlen, 1, 1909, 545-505. 



97 Jour. Indian Math. Club, 2, 1910, 202. 



98 L'intermSdiaire des math., 17, 1910, 288. On pp. 72, 166 it is shown that his earlier solu- 



tion, 16, 1909, 220, is not general. 



"Ibid., 18, 1911, 62. Gleizes, 21, 1914, 156-7, stated that we may need to give fractional 

 values to I, m, n, p, v. 



100 Ibid., 17, 1910, 278; Sphinx-Oedipe, 1907-8, 27. 



101 Sphinx-Oedipe, 6, 1911, 25-26. Proposed bv F. Proth, Nouv. Corresp. Math., 4, 1878, 95. 



