CHAP. VI] SUM OF TWO SQUARES. 243 



P. Seeling 92 proved that if A is a prime 4m + 1 the period of the con- 

 tinued fraction for VA has an odd number of terms. Hence A is a EC. 



J. Petersen 93 reproduced Euler's 24 proof that every divisor of a sum of 

 two relatively prime squares is a El. Then by Wilson's theorem, every 

 prime 4n + 1 is a EL He proved Gauss' 37 result on the number of solu- 

 tions of z 2 + 2/ 2 = A. 



L. Lorenz 94 proved that 



--OO 00 00 



m, n= oo m=0 n=l 



whence m? + n* = N has 4(a# 6^) solutions if a N is the number of 

 divisors of the form 4m + 1 of N, and b N the number of divisors of the 

 form. 4m + 3. 



P. Bachmann 95 employed the theory of roots of unity to prove that 

 every prime p = 4n + 1 is a sum of two squares, to compute the squares, 

 and to prove Gauss' 44 result. 



J. W. L. Glaisher 96 would strike out of the list of numbers 



123456--. 

 -3 -6 -9 -12 -15 -18 



5 10 15 20 25 30 

 - 7 - 14 - 21 - 28 - 35 - 42 



9 18 27 36 45 54 



every one whose negative occurs in the list. Each remaining positive 

 number 1, 2, 4, 5, 8, 9, 10, is a El and every 121 occurs in the final set. 

 The proof is by Jacobi's 46 formula. He gave a like scheme to obtain the 

 numbers expressible as a sum of two odd squares. 



R. Hoppe 97 proved that every prime p = 4n + 1 is a El. The values 

 of r = x 2 for x = 1, , 2n are incongruent modulo p. But r 2n = 1 has 

 only 2n roots and r is a root. Hence to each x corresponds an integer y 

 such that y 2 = r. Thus x 2 + y* = pq. If p\ is a factor of q, we get 

 3? + 2/i = Piqi- Since the q's decrease, we finally get a q k = 1, whence 

 %l + yl = Pk- The remaining factors of qk-i are C2, whence qk-i is a El. 

 Then pk-i = EO /<?*_! = El, etc. Finally, p is a EL 



F. L. F. Chavannes 98 considered an integer N whose prime factors are 

 distinct and each of the form 4e + 1 and hence a El. Thus N = II (a 2 + /3 2 ). 

 Set Ni = (a 2 + /3 2 )( T 2 + 5 2 ), N* = N^ 2 + r 2 ), , whence NI = x\ + yl 

 for xi = 0:7 j85, yi = (3y ^ <* 5 - Similarly, each pair Xi, y\ yields two 



92 Archiv Math. Phys., 52, 1871, 40-9. 



93 Tidsskrift for Math., (3), 1, 1871, 80-4. 

 M Ibid., 97. 



96 Die Lehre von der Kreistheilung, 1872, 122-137, 235. 



96 Math. Quest. Educ. Times, 20, 1873, 87; British Assoc. Report, 46, 1873, 10-12 (Trans. 



Sect.). 



97 Archiv Math. Phys., 56, 1874, 223. 



98 Bull. Soc. Vaudoise des Sc. Naturelles, Lausanne, 13, 1874-5, 477-509. 



