274 HISTORY OF THE THEORY OF NUMBERS. [CHAP, vii 



W. C. Eells, 109 to solve x 2 + y 2 + z 2 = a 2 , took x = 2MN, y = M 2 - N 2 , 

 a = m 2 + n 2 , and gave to M 2 + N 2 , z the values m 2 w 2 , 2mw in either 

 order. He tabulated 125 sets of solutions arranged according to the size of a. 



A. Ge"rardin and E. Miot 110 gave many identities x 2 + y 2 = u 2 + v 2 + w 2 . 



L. Aubry 111 gave a very long, but elementary proof, by use of theorems 

 on divisors of numbers x 2 + my 2 , that every number not of the form 

 4 r (8n + 7) is a GO. 



L. J. Mordell 112 proved Kronecker's 33 theorem by use of theta functions. 



A. S. Werebrusow 113 noted that the problem to find two equal sums of 

 three squares is evidently equivalent to mm' + nn' + pp' = 0, the general 

 solution of which is stated to be 



m = aft ba, n = ay ca, p = a8 dot, 

 m f = cd dy } n' = dp bd, p f = by c0. 



He gave long formulas said to solve x 2 + y 2 = u 2 + v 2 + w 2 completely. 



E. Bahier 114 found solutions of a 2 + b 2 + c 2 = d 2 in which a + b = d, 

 d = c + 1, d 2 = c 2 + 7 2 > or a and b are given. He discussed the nature of 

 numbers d such that d 2 is a sum of three squares =1= 0. 



E. Turriere 115 derived (4) geometrically and showed how to deduce new 

 solutions of x\ -\- + x\ = R 2 from a given solution. 



W. de Tannenberg 116 found real polynomials of degree 2n in a variable 6 

 satisfying x 2 + y 2 + z 2 = P 2 , where P is a given polynomial of degree 2n in 6, 

 not zero for any real 0. Hence set P = (a\ t\] (al Q,t p = i(0-{- b p ). 

 For arbitrary parameters a , , <x n , define two sets of functions by 



Up = (dpUp-i + tpVp-^e^, v p = (dpVp-i + t p Up-i)e~ iap (p = 1, , n), 



UQ = e ta , v = e~ ia . Let the u, v become u', v' when ti, -,t n are changed 

 in sign. Define x, y, z by means of 



P z = 2u n Vn, P + z = 2v n u' n , x + iy = 2u n u' n , x iy = 2v n v' n , 



which are consistent since u n v' n + v n u' n P. Take t p = i(6 + b p ). 



On two equal sums of three squares, see papers 19 and 86 of Ch. VIII. 

 By Cesaro 26 of Ch. IX there are in mean ^irn 112 representations of n as a SL 

 On a m equal to 2v 2 , v 2 or v 4 , see papers 171 of Ch. XIII, 69 of Ch. XV, 312 

 of Ch. XXII. On numbers not a OS, papers 4, 5 of Ch. VIII. On systems 

 of equations including m = D, papers 97 of Ch. VII, 94 of Ch. IX, 32-39a, 

 51, 146, 165, 168 of Ch. XIX, 390-8 of Ch. XXI, 308-9 of Ch. XXII. On 

 systems including m = u 9 or w 5 , papers 95, 97 of Ch. XX, 353, 392, 402-3 

 of Ch. XXI. 



109 Amer. Math. Monthly, 21, 1914, 269-273. 



110 L'interm4diaire des math., 21, 1914, 190-2. 



111 Sphinx-Oedipe, num6ro spdcial, Jan., 1914, 1-24. 



112 Mess. Math., 45, 1915, 78. 



113 L'interm<5diaire des math., 23, 1916, 12-13, 17-18. 



114 Recherche . . . Triangles Rectangles en Nombres Entiers, 1916, 234-254. 

 116 L'enseignement math., 18, 1918, 90-5. 



116 Comptes Rendus Paris, 165, 1917, 783-i. 



