252 Mr Morddl, On the representation of algebraic numbers 



if a is positive and the discriminant iac - b^ is neither negative nor 

 of the form M. 



It will be useful hereafter to prove now that if q is any integer, 

 an even integer p can be found so that q — p^ is not of the form M. 

 This follows from (A), for if 



q = 1,2 (mod 4), take p =0 (mod 2), 

 if ^ = 3 (mod 8), take 2? = (mod 4), 



if q = 7 (mod 8), take p = 2 (mod 4). 



If however q = (mod 4) 

 put ? = 4^i>P = 2pi; 



then since 



q-p^=4:{qj^- 2\^) 



the result follows by induction since if 



5-1=0 (mod 4), 

 we can put q^ = 4^2, Pi = '^p^, etc. 



§ (3). Hubert's problem can be simplified by writing 



y = Ax^ + Bx+ C 



so that X can also be expressed rationally in terms of y by aid of 

 equation (1), while y is also the root of an irreducible cubic*. Hence 

 the whole question can be reduced to proving H for x where 



x^ — ax^ -\- bx — c ^ (1) 



where the real roots of the cubic and hence also c are positive. The 

 cubic in x can be written as 



{a -]- p) x^ + {q — b) X + c 

 x^ + px + q 



where p and q are entirely arbitrary. If we can find rational values 

 for p and q so that the discriminants 



p = iq — p^ and a = ic {a -\- p) — {q — b)^ (3) 



are both positive and neither of them of the form M, then H holds 

 for both the numerator and denominator of x and hence for x. 



Suppose now that a, b and c are all positive, including in par- 

 ticular the case when all the values of x are real. Then by con 

 sidering the region common to the two parabolas 



4rj - ^2 _ and 4c (« + I) - (77 - bf = (4) 



it is obvious that real and hence also rational values of p and q can 

 be found to make each of the discriminants in (3) positive. Moreover, 

 if we can find any rational valuesf of p and q for which neither of the 



* Except in the trivial case A = B = 0. 



f It is by no means obvious that such values exist. For example, one of the two 

 expressions, k + 1, 512 - k^ is always of the form M. The same applies to the two 

 expressions pq, 512g'2 - (^7q - p)^, 



_\u -^ jj) jb- -r \i[ — uj jy -r V -f). 



