CHAP. VI] SUM OF Two SQUARES. 251 



6 > a > 0, then S<r(p) = 48N (mod 32). For p = 4n + 1, 



<r(p) = 2(2n + 1) 

 and N is odd. 



A. Matrot 134 noted that, if p = 2h + 1 is a prime, and a is not divisible 

 by p, a h = 1 (mod p) by Fermat's theorem. If the upper sign held for 

 every a, 



s h = l h + + (p - l) h = p - 1 (mod p), 



whereas, for # < p 1, s 3 = (mod p}, as shown by induction. Hence 

 there exists an a for which a h = 1. Let h = 2k. Thus p divides a EL 

 That p is a El follows as in his 1891 paper on ID. 



E. Catalan 135 repeated the proof by Eugenio. 91 



H. Weber 136 proved that every prime n = 4f + 1 is a E] by use of the 

 four periods each of / terms of nth roots of unity. 



C. Stormer 137 proved that 1 + x 2 4= 2y n if x \ > 1 and n has an odd 

 divisor > 1. 

 '"Several 138 treated x 2 + (x + I) 2 = y\ whence P - 2u 2 = - 1 if 



t = 2x + 1, u = y 2 . 



Stormer 139 applied a theorem on Pell's equation (Stormer 230 of Ch. XII) 

 to find the complete solution of 1 + x 2 = kA? - A" n n in positive integers, 

 where k, AI, , A n are given positive integers. In particular, there is a 

 new proof that 1 + x 2 = y n or 2y n is impossible if z > l,y > 1, n being an 

 odd prime. 



M. A. Gruber 140 gave a table and identities f or 4n + 1 = EL 



Several writers 141 discussed x 2 + p 2 = y 3 for p a prime. 



G. de Longchamps 142 noted that N 4 is a GO or SI if N/\ 1 is a square 

 or El, since 



N 4 = 1Q\(N -\)(N - 2X) 2 + (N 2 - 8\N + 8X 2 ) 2 . 



R. Hoppe 143 used Girard's theorem to prove that a number is a El or 

 not according as it has no prune factor of the form 4n 1 to an odd power 

 or at least one such prime power factor. 



J. H. McDonald 144 gave a direct proof of Jacobi's 48 result on the number 

 of representations of an odd positive number as a E] . 



C. A. Laisant 145 noted that (a 4n+2 + l)/(a 2 + 1) is always a El. 



134 Jour, de math. 616m., (4), 2, 1893, 73. 



136 M6m. Acad. Roy. Belgique, 52, 1893-4, 17. 



136 Lehrbuch der Algebra, I, 1895, 583-5; ed. 2, I, 1898, 632-4. 



137 L'interm^diaire des math., 3, 1896, 171; 5, 1898, 94 for n = 2 m . 



138 Ibid., 4, 1897, 212-5. 



139 Videnskabs-Selskabets Skrifter, Christiania, 1897, No. 2. 



140 Amer. Math. Monthly, 5, 1898, 240-3. 



141 L'intermediaire des math., 5, 1898, 157-9; 16, 1909, 177. 



142 Ibid., 7, 1900, 65. Misprint of 2N - \ for N - 2X. 



143 Archiv Math. Phys., (2), 17, 1900, 128, 333. 



144 Proc. and Trans. Roy. Soc. Canada, (2), 6, 1900, Sec. Ill, 77-8. 

 146 Nouv. Ann. Math., (4), 1, 1901, 239-240. 



