302 HISTORY OF THE THEORY OF NUMBERS. [CHAP, vm 



A. Martin 102 noted that, if t = 2p 2 + 2q 2 - n 2 , the sum of the squares 

 of t + 4np, t 4np, t + 4nq, t 4n# equals the square of 4p 2 + 4q 2 + 2n 2 . 

 Also [Aida 59 of Ch. IX], 



(p 2 + q* + r 2 - s 2 ) 2 + (2ps) 2 + (2gs) 2 + (2rs) 2 = (p 2 + g 2 + r 2 + s 2 ) 2 . 



P. Bachmann 103 gave an exposition of papers by Glaisher, 70 Dirichlet, 42 

 and Stern. 81 



L. Aubry 104 proved that every integer N is a HL It evidently suffices 

 to treat the case N odd or double an odd number. It is first shown that 

 N divides a certain X 2 + Y 2 + 1, where we may take X ^ NJ2, Y ^ AT/2. 

 Consider therefore the numbers NI, N 2 , defined by 



XI + Y\ + 1 = NtNn-1, Xi ^ NJ2, Y, ^ N,J2. 

 The N's form a decreasing series of positive integers. Hence a certain N n 

 is unity. Then N n ^ = X*_i + H^i + 1. But if 



X2 + 72 + i = DE, E = p 2 + q 2 + r 2 + s 2 , - pX + rF + s = aE, 

 sX + qY + p = cE, qX - sY + r = dE, rX + pY - q = bE, 



then D = a 2 + 6 2 + c 2 + d 2 . Applying this theorem for p = 1, r = 0, 

 s = Xn-i, q = Y n -i, X = Z n _ 2 , Y = Fn-2, whence D = N n _ 2 , -& = A^ n _i, 

 we see that N n -z is a HJ . By the same theorem we see by induction that 

 every Ni is a 3] . Hence N = NI is a 3D . [There is no explicit proof that 

 a, - , d may be taken to be integers and hence that the decomposition is 

 not merely into four rational squares.] 



E. Dubouis 105 proved that Descartes' 5 statements are true. The 

 numbers not a sum of 4 squares > are 1, 3, 5, 9, 11, 17, 29, 41 and 

 4"X (X = 2, 6, 14), n ^ 0. 



S. A. Corey 106 gave a vector interpretation of (1) by use of four pentagons 

 with a common vertex and four consecutive sides in one pentagon parallel 

 to corresponding sides of the others. 



C. van E. Tengbergen 107 proved that x z + y 2 + z z = (mod p} has 

 (p l)(p &)/48 sets of solutions < p/2, where k = 1, 5, 11, 17 

 according as the prime p = 8v 1,80 3, 8v + 3, Sv + 1. 



E. Landau 108 proved that the number of sets of integral solutions of 

 M a + t>* + t* + s/* < 3 is iirV + 0(z 1+ '), for e > and as by Landau, 179 

 Ch. VI. 



G. Metrod 109 solved x 2 + (a; + yY + (x + 2y) 2 + (x + 3i/) 2 = z 2 for x; 

 the radical is rational if z 2 5y 2 = u z and hence if z = a 2 + 56 2 , 2/ = 2a&, 

 w = a 2 56 2 . 



L. Aubry 110 showed how to find all solutions of a 2 + b 2 + c 2 + d 2 = N, 

 first when a 2 + b 2 , ac + 6d and N have no common factor, and next when 



102 Amer. Math. Monthly, 16, 1909, 19-20. 



103 Niedere Zahlentheorie, 2, 1910, 286, 323, 348-358. 



104 Assoc. frang., 40, 1911, I, 61-6. 



105 L'intermddiaire des math., 18, 1911, 55-6, 224-5. 



106 Amer. Math. Monthly, 18, 1911, 183. 



107 Wiekundige Opgaven, Amsterdam, 11, 1913, 244-7. 



108 Gottingen Nachrichten, 1912, 765-6. 



109 Sphinx-Oedipe, 8, 1913, 129-130. 



110 Ibid., num6ro special, March, 1914, 1-14; errata, 39. 



