From: cabay@cs.ualberta.ca (Stan Cabay)
Subject: Number Theory
Originator: dan@symcom.math.uiuc.edu
X-Submissions-To: sci-math-research@uiuc.edu
Organization: University of Alberta, Edmonton, Canada
X-Administrivia-To: sci-math-research-request@uiuc.edu
Date: Mon, 5 Oct 1992 17:59:37 GMT

 A student here has some results in number theory which we are
 having difficulty in evaluating. I would like any comments
 on the results discussed in the following abstract with regards
 to new, worthwhile, etc.  What are appropriate references?
 
 Stan Cabay
 Associate Professor
 
 /* Abstract */


  - Functional form invariance plays a key role in number theory.
 Composition laws of the Binary Quadratic forms expresses this with
 simplest important example coming  from Diophantus sum of
 squares. The continued fraction expansion for N gives the
 well known algorithm for irrational integers. 
 The solution of Fermat's equation,

         (-1)^n  = P(n)^2 - N*Q(n)^2,   

 follows from this algorithm.  From a generalization of Diophantus equation 
 (due to Euler) in the form of a sum of 4-squares and of a sum of 8-squares 
 (by Caley), we prove the existence of infinite solutions for the 

  4-D case :

    [1]       (-1)^n = X(n)^2 + Y(n)^2 - N*( W(n)^2 +Z(n)^2)

 and the 8-D case :

    [2]    (-1)^n = (R(n)^2 + T(n)^2 + X(n)^2 + Y(n)^2) - 
                N*(U(n)^2 + V(n)^2 + W(n)^2 + Z(n)^2)

 for any odd prime N. Generalizing further on the number of variables and 
 using an application of Hurtwiz-Radon theorem, we prove  that all odd 
 square-free  integers > 1 admit a solution to a generalized Fermat equation 
                   
         (-1)^n  = |P(n)|^2 - N*|Q(n)|^2,            |a| = modulus of a,

 where  P(n), Q(n) are VECTOR convergents  of order n,
 with dimension D=u*2^(4a+b) where  a>0,  0<=b<=3, u >1 natural numbers        
 (this theorem is basically Radon's; Hurtwiz gave the D=2,4,8 cases
 to be found on 
    Hurtwiz - "Uber die Komposition  der quadratischen Formen",
               Math. Werke, Bd. II, 565-571.
    Radon   -  "Lineare Scharen orthogonaler Matrizen", Math. Werke.  )

   The existence of the general solution, was found to be numerically 
 independent of the parity of n (n odd or even), i.e. the negative 
 unit solution did not discriminate on the representation of N as 
 sum of K= D/2 +1 (D>2) NON VANISHING squares. The case D=2 is very peculiar
 since the number of representations for K=2 sum of squares is exactely 1 
 when N is 1 mod 4, and only then the solution for n-odd exists.

 For D=4*p, p>0, the Hurtwiz theorem gives the conditions for which 
 the equation below

 (x1^2 + X2^2 + ... XD^2) = (y1^2 + y2^2 + ... yD^2)*(z1^2 + z2^2 + ... zD^2)

 can hold in the integer field.  This equation is clearly a generalized
 form of the Diophantus equation for a sum of 2-squares.
 When D=4, the algorithm shows that N can in general be written as 
 the modulus of a complex continued fraction 
                                                     
     |  A0+iB0 + 1                                  | 2  
     |          ---------------	                    |
     |          A1-iB1 + 1                          |  
 N = |                  -------                     |
     |                  A2+iB2 + 1                  |   
     |                           ---------          |
     |                           A3-iB3  + ....     |
  

 In short hand notation S(n) = (A(n) +iB(n)) + 1
                                             ------    ~=conjugate number
                                             ~S(n+1) 
                                                          
 is the continued fraction, and its square modulus gives N; in the case D=8 one 
 would utilize quaternions and take the square norm of a 4-vector.
                             
 At the moment, the early results seem to indicate continued fractions 
 with substantial differences compared with the 2-D case.
 The major problem i've encountered lies at the heart of this representa-
 tion as sum of K squares. When we have 2-dimensions the problem is indeed well 
 defined: the representation as sum of 2 squares (for a prime) is unique, 
 this does not happen in higher dimensions (any prime = sum of three squares
 has a very large representations as the size grows!). This latter 
 fact seems to influence the solutions of [1] and [2] very deeply.

 For example : take N =  431  ->  3 mod 4 and -1 mod 8
 this number has NO representation as sum of 2 and 3-squares, yet 
 equation [1] has the solution
                            
                      -1 = 69^2 +29^2  - 431*(3^2 +2^2)

 and all its odd powers (like it's done in the 2-Dim case).
 (the corresponding 2-Dim solution does not exist e.g. is not 1 mod 4)
 An approximate expansion for  N is

          |  4-6i + 1                    |2
          |        ------                |  
    431 = |         1+i + 1              |
          |              ----            |
          |              4 -2i +....     |                           
 

 My results do not seem to explain why [1] is satisfied for n-odd and N does
 not have representations as sum of K=3 squares, nor have I
 a well formed explanation for this. The 8-D case follows the same 
 behavioural path. The periodicity of the fraction seems also to be happening poorly,
 I've got some expansions that do not show periodicity at all
 (i.e., terminated expansion because A(n)=B(n) = 0) and some that do. I suspect
 that this is tied with these multiple representations and to the value K
 which is always odd for D >2 (even). In other words, for a sum of K-(odd) 
 squares the corresponding D-dimensional Fermat equation always has the 
 negative unit solution (means that  -1 is on the LHS of the equation), 
 a fact that I find very surprising.

 To make K-even is to make D=2*(odd number) and such possibility leads us 
 to say immediately this still gives the same result as the K= odd cases 
 because for example, in the case D=6, the product of a sum of 6-squares 
 can be only written as sum of 8-squares which brings us back to K+1=5,
 this in turn implies those solution characteristics. Does this then mean that
 the negative solution exists for ALL K's >2 ? (i.e. K basically doesn't 
 matter!). Well, for D>6 The answer (it feels that way) seems to be yes. 


Newsgroups: sci.math.research