Conference rusure::math

1692.0. "American Math Monthly 10258" by RUSURE::EDP (Always mount a scratch monkey.) Mon Nov 09 1992 14:02

    Proposed by Hans Liebeck and Anthony Osborne, University of Keele,
    England.
    
    Let a, b, and c be positive integers which are pairwise relatively
    prime.  Prove that if the congruences
    
    	A^2 = -bc (mod a), B^2 = -ca (mod b), C^2 = -ab (mod c)
    
    are solvable for A, B and C, then the equation
    
    	ax^2 + by^2 + cz^2 = abc
    
    has a solution in integers x, y, and z.

    Proposed by Jonathan L. King, University of Florida, Gainesville, FL.
    
    Let <r[k]> for k in the natural numbers be defined by r[0] = 3 and
    r[k+1] = r[k]^2-2.  Evaluate
    
    	limit as K goes to infinity of
    
    		[product from k=0 to K-1 of r[k]] ^ [1/2^K].

    Note 1692.1 should have been its own topic.  But as it turns out, it is
    related to 1668 and is answered there.
    
    
    				-- edp