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



Set a = = p = 1, a' = - = <?' = 1. Then a factor 4m + 1 is 

 replaced by + 1, a factor 4m + 3 by 1. Hence the product is replaced 

 by E. Thus 



Hence E = unless A', , S f are all even. If the latter are all even, E 

 is the number of factors of n = a A - - p R , while p = nQ 2 , where every prime 

 factor of Q is of the form 4m -f 3. Now 2p is not a 03 unless p is of this 

 form nQ 2 . Also 2nQ 2 = y 2 + 2 2 requires that ?/ and 2 be divisible by Q, 

 while 2n = w 2 + x 2 has as many sets of positive solutions as n has factors 

 (all the factors of n being of the form 4m +1). 



A. D. Wheeler 51 gave trivial or known results on . 



G. L. Dirichlet 52 obtained, as a special case of a general thoerem on 

 quadratic forms, Jacobi's 48 result that, if n is odd and positive, the number 

 of sets of solutions of x 2 + y 2 = n is the quadruple of the excess of the 

 number of divisors 4k + 1 of n over the number of divisors 4& + 3. 



A. Cauchy 53 proved Gauss' 44 result that, if p = x 2 + y 2 , 



Cauchy 54 proved identities of the type 

 + 2t + 2t* + 2t + - 2 = 1 + 2Z 2 + 2t* 



G. Eisenstein 55 gave the values of A, B in p = 4n + 1 = A 2 + B 2 and 

 p = 3n + l = A 2 A.B -f B 2 , where p is a prime. He 56 stated that the 

 number of lattice points inside and on the circumference of a circle of radius 

 Vm and center at the origin is 



C. G. J. Jacobi 57 gave the representation as a [2 of each prime 

 4n + 1 ^ 11981. 



Jacobi 1 noted in 1847 that an insignificant change in the text of 

 Diophantus V, 12 gives the result that, if a number without a square 

 factor is a El, neither itself nor a factor of it has the form 4n 1, and 

 expressed his belief that Diophantus had a proof, though he gave none, 

 since all that is essential to a proof was in the Greek mathematics and is 



61 Amer. Jour. Sc. and Arts (ed., B. Silliman), 25, 1834, 87. 



Jour, fur Math., 21, 1840, 3; Werke, I, 463. Zahlentheorie, 91. 



M M6m. Ac. Sc. Paris, 17, 1840, 726; Oeuvres, (1), 3, 1911, 414. 



84 Comptes Rendus Paris, 17, 1843, 523, 567; Oeuvres, (1), VIII, 50, 54. 



66 Jour, fur Math., 27, 1844, 274. 



K Ibid,, 28, 1844, 248. Cf. Gauss, 46 Suhle 730 and Cayley. 81 Proved also by H. Ahlborn, 



Ueber Berechnung von Summen von grossten Ganzen auf geometrischem Wege, Progr. 



Hamburg, 1881, 18. 

 M Jour, fur Math., 30, 1846, 174-6; Werke, VI, 265-7. Errata, Mess. Math., 34, 1904, 132. 



