CHAP. VI] SUM OF TWO SQUARES. 255 



Marchand 174 presented the known application of complex integers 

 a -f bi to find all decompositions of a product of primes 4n + 1 as a GO. 



Paulmier 175 gave solutions of x 2 + y 2 = A 3 for five special values of A. 



Several writers 176 found x such that x + 1 and x 2 -f 2 are sums of two 

 squares. 



J. K. Heydon 177 noted that, if a, b, are distinct primes, 



a v - l b*'- 1 - - = El in 2*++ 1 or ways. 



P. Lambert 178 applied complex integers a + bi. He gave two proofs 

 that a divisor of a CU is a OH . 



E. Landau 179 proved that, if A(x) is the number of pairs of integers 

 u, v for which u 2 + v 2 ^ x, then A(x) = TTX + 0(o: 1/3+ ), for every e > 0. 

 Here/(:c) = 0(g(x}) means a function such that there exist two numbers 

 and A for which | f(x) < Ag(x) when a; ^ . Although the result is 

 not quite as sharp as that by Sierpinski, 155 the proof is much shorter. 



Landau 180 gave a new proof of the theorem due to Sierpinski. 155 

 R. Bricard 181 gave an elementary proof that every prime p = 4n + 1 

 is a E]. By Wilson's theorem, m 2 + 1 = (mod p) for m = [(p l)/2j!. 

 Write Xi for the minimum residue of mi modulo p. Consider the p 1 

 points Mi = (x{, i). The square of the distance M^Mj between any two 

 of these points is divisible by p. It is shown that the least of these squares 

 is < 2p if p > 32 and hence equals p. A like proof shows that every prime 

 Sq 1 is of the form x* - 2y z . 



F. Ferrari 182 noted that the least number decomposable in 2 n distinct 

 ways as a sum of two relatively prime squares 4= is the product, found by 

 (1), of the first n + 1 prunes of the form 4& + 1. For this least x = p 2 . + q] 

 (i = 1, , 2 n ), set y f = p] q], z f = 2piqi] then z 2 = y\ -f- z\ is the least 

 square decomposable hi 2" ways as a El. To find the least (p + l)th 

 power decomposable hi 2 P ways as a El, use P = 6* + c* (i = 1, , 2 P ), 

 whence E(6j + cj) = P p+l has 2 P decompositions. 



A. Aubry 183 noted that (1) can be derived from Brahmegupta's (Ch. V) 

 inscribed quadrilateral A BCD whose diagonals meet at right angles at 0, 

 by evaluating the perpendiculars BE and OJ to DC. 



E. Haentzschel 184 noted that his 152 method in Ch. XXI to deduce a new 

 solution of 'ax 3 + + d = y 3 from one solution may be applied to 

 x 2 + y 2 = z 3 in two ways according as x or y is taken as the variable. He 



174 L'intermediaire des math., 18, 1911, 228-232. 

 176 Ibid., 19, 1912, 151. 



176 Ibid., 55-7, 257. 



177 Math. Quest. Educ. Times, (2), 21, 1912, 98-9. 



178 Nouv. Ann. Math., (4), 12, 1912, 408-421. 



179 Gottingen Nachrichten, 1912, 691-2. Giornale di Mat., 51, 1913, 73-81. 



180 Annali di Mat., (3), 20, 1913, 1-28; Sitzungsber. Akad. Wiss. Wien (Math.), 121, 1912, 

 Ha, 2298-2328. 



181 Nouv. Ann. Math., (4), 13, 1913, 558-562. 



182 Periodico di Mat., 28, 1913, 71-8. 



183 Sphinx-Oedipe, numero special, June, 1913, 23-24. 



184 Sitzungsber. Berlin Math. Gesell., 13, 1914, 92-6. 



