254 HISTORY OF THE THEORY OF NUMBERS. [CHAP, vi 



F. Ferrari 164 found the known solution of z 2 + y 2 = z n by use of 

 z = r + si. 



H. Brocard 165 noted that ri 2 + (n + I) 2 = m k has solutions for k = 2, 

 but not for k = 3. 



E. Landau 166 considered the number B(x) of positive integers ^ x 

 which are 03 and gave a long proof that 



B(x) Viogz 1 / / n IV 1 

 =-=<v/n( 1--=) , 

 \2 x \ r 1 / 



*= x 



where r ranges over all primes of the form 4m + 3. 



E. Landau 167 applied binary quadratic forms to show that a number is 

 a E] if and only if it has no prime factor 4m -f- 3 to an odd power. 



E. N. Barisien 168 used the epicycloid to derive the identity 



(8Z 3 - Qt 2 - Qt + 3) 2 + 4(1 - Z 2 )(l + 3Z - 4i 2 ) 2 = 13 - 12$, 



whence 12 - - 13Hs a m if t = (1 - 2 )/(l + 2 ). 



M. Kaba and L. E. Dickson 169 deduced, by use of special theta functions, 



Hence there is no representation as a [21 of a number having a prime factor 

 4m + 3 with an odd exponent, and no proper representation when such a 

 factor has an even exponent. If P = p" 1 - p* 1 , where pi, , p, are all 

 the distinct primes of the form 4m + 3 which divide e, and if x b , TT S 

 are all even, there are as many improper representations of e as there are 

 representations of e/P', every representation of e is of the type (P 1 / 2 ^) 2 

 _|_ (pi/2^)2 Hence the problem reduces to the case in which every prime 

 factor of e is of the form 4m + 1. Then the number of representations of e 

 as a 12] is (TTI + 1) (ir n + 1). 



P. Bachmann 170 gave an exposition of the work of Lagrange 32 and 

 Vahlen. 133 



Welsch 171 stated that the general solution of u? + x 2 = y 2 + z 2 is 



2x = ab + cd, 2y = ac + bd, 2z = db cd, 2u = ac bd, 



where a, d are even, or b, c are even, or all four are odd. 



L. Aubry 172 proved that x 2 + (x + I) 2 + m k if k is not a power of 2. 



A. Deltour 173 applied continuants (Muir 101 ) to prove that a prime 

 4/z, + 1 is a -12] in one and but one way. 



164 Periodico di Mat., 25, 1909-10, 59-66; Supplem. al Period, di Mat., 12, 1908-9, 132-4. 

 166 L'interm6diaire des math., 15, 1908, 18-19. 



166 Archiv Math. Phys., (3), 13, 1908, 305-12. 



167 Handbuch . . . Verteilung der Primzahlen, 1, 1909, 549-550. 



168 Assoc. frang. av. sc., 38, 1909, 101-7. 

 189 Amer. Math. Monthly, 16, 1909, 85-7. 



170 Niedere Zahlentheoric, 2, 1910, 304-319 (477). 



171 L'interm6diaire des math., 17, 1910, 96, 118, 205. 



172 Ibid., 18, 1911, 8-9; errata, 113; Sphinx-Oedipe, num6ro special, March, 1914, 15-16; 



errata, 39. 



173 Nouv. Ann. Math., (4), 11, 1911, 116. 



