CHAP. VI] SUM OF TWO SQUARES. 241 



For p a given integer, let ni, , /i s denote the integers prime to p and < p. 

 In the continued fraction for p/nk, [_q\ q n ~] is now p. In view of (2), 

 [?n q\] arises from some p//ji k ,. Let p be a prime 4X + 1, so that 

 s = 2X. Hence there is some nk =J= 1 which coincides with \i k , and thus 

 there is a set of quotients q lt - , q n symmetrical from the ends. If n 

 were odd, n = 2i 1 ^ 3, p = [g x g<_ig#f_i gi] has the factor 

 <?t-i] by (3). Hence n = 2i and 



C. G. Reuschle 80 expressed as a sum of two squares each prime 4n + 1 

 up to 12377, and to 24917 for those primes for which 10 is a quadratic 

 residue. 



A. Cayley 81 wrote E'(n/K) = 1 or according as n(k is an integer or not 

 and proved that the number of ways the integer n is a C3 is 



v = E'(n) - E'(n/3) + E'(n/&) - E'(n/7) + -, 



if n = a 2 + /3 2 is counted twice when a 4= Hence v is the number of 

 lattice points on the quadrant of the circle with radius Vw and center at 

 the origin. Eisenstein's 56 formula follows readily. 

 J. Liouville 82 stated the formula 



-9"], 



summed for s = 1, 3, 5, and for 6 = 0, 1, 2, , [V], and implied 

 that it is connected with sums of two squares. It was proved geometrically 

 by L. Goldschmidt, 83 who showed that the right member is the number of 

 lattice points in a quadrant of the circle r 2 + 2 = n. 



F. Unferdinger 84 proved, by use of norms of complex numbers, that a 

 product of n sums of two squares can be expressed as a (2] in 2 " 1 ways, 

 distinct in general. 



S. Kaminsky 85 proved that x 2 + y 2 = pz 2 is impossible in integers if p 

 is a prime 4n + 3. 



F. Woepcke 86 proved by induction from p, p n , p n+1 to p n+z that any 

 power of a prime 4m + 1 can be expressed in one and but one way as a 

 sum of two relatively prime squares. The proof shows that the number 

 of all decompositions (primitive or not) of p A as a C2 is (X + l)/2 if p is 

 odd, X/2 if p = 2. Hence follows Gauss' 37 formula. Also the number 

 of primitive decompositions of pT Pl v is 2"" 1 , if each pi is of the form 

 4m + 1. 



J. Plana 87 used Jacobi's 46 formula to prove Gauss' 37 result on the number 

 of ways of expressing N = 2 li S z p a p' ? as a 2 + 6 2 , where p, p', - are 



80 Math. Abh., Neue Zahlenth. Tabellen, Progr. Stuttgart, 1856. Errata by Cunningham, 



Mess. Math., 34, 1904-5, 133-5. 



81 Quar. Jour. Math., 1, 1857, 186-191. 



82 Jour, de Math., (2), 5, 1860, 287-8. 



83 Beitrage zur Theorie der quad. Formen, Diss. Gottingen, Sondershausen, 1881. 



84 Archiv Math. Phys., 34, 1860, 83-100. 



85 Nouv. Ann. Math., (1), 20, 1861, 97-9. 



88 Atti Accad. Pont. Nuovi Lincei, 14, 1860-1, 311-5. 



87 Mem. Accad. Turin, (2), 20, 1863, 123-6. 



17 



