10 



REPORT 1873. 



On certain Propositions in the Theory of Numbers deduced from Ellijitic- 

 transcendent Identities. By J. W. L. Glaishek, B.A. 



The paper consisted of a series of propositions in the theory of niimhers deduced 

 from identities either actually or implicitly g^iven in Jacobi's ' Fundamenta Nova ' 

 (Regiomonti, 1829), and of which the author believed some might be new. In this 

 abstract the demonstrations are omitted, and only the enunciations of the proposi- 

 tions, with one or two examples of each, are given. 



(i) Construct the following scheme : — 



the mode of formation of which is evident ; then strike out all the numbers that 

 cancel one another, and every number that remains is either a square or is expres- 

 sible as the sum of two squares ; the converse proposition, that every number that 

 is a square or is expressible as the sum of two squares will remain, is also true. 

 Thus, 1 = 1-, 2=l--t-l^ 3 is cancelled, 4=2", 5=2^ 4-1^ 6 and 7 are cancelled, 

 8=2^+2^, 9 is cancelled in the 3-line, but reappears in the 9-line, so that it re- 

 mains as it ought to do, since it=:3^, 10 = 3^-1-1-, 11 and 12 are cancelled, &c. 



(ii) Every number which is a square or expressible as the simi of two squares is 

 of the form 2'(4m— 1)-«, n being any odd number, all of whose factors are of the 

 form 4a+l, and I and m any positive numbers ; and if \|/'(«) denote the numbers of 

 factors of 71 (unity and n itself included), then the number of ways in which 

 2'(4»i — l)^w can be expressed as the sum of two squares =h'^{n); but if the 

 number be a square, or the double of a square, the number of ways 



=i{V.(M)-l} or i{t/.(«)+l} 



respectively (0^ not being counted as a square). From this many well-known 

 theorems follow at once. 



(iii) The following is the " sieve " corresponding to that in (i) for numbers that 

 are the sum of two odd squares. 



Every number that remains after the cancelling is the sum of two odd squares ; and, 

 vice versa, every such number remains. 



(iv) Every number that is the sum of two odd squares is of the form 2(4m— 1)2« ; 

 and ever}' such number can be expressed as the sum of two odd squares in \^{n) 

 ways, unless it is the double of a square, when, if of the form 2(4wi— 1)-, it cannot 



