as a sum of four squares 253 



discriminants in (3) is of the form M, it is clear that the same holds 

 when f and q are replaced by 



if N is sufficiently large and A, fx are fractions with odd denomina- 

 tors, and that A, /x, N can be taken so as to bring the point 



(p + 2^A, q + 2%) 



within the common region of the parabolas (4), that is, p and a 

 will be both positive. 



We can summarise the rest of this section § (3) by saying that 

 we shall show that suitable values for p and q can be found when 

 either a or 6 or c is odd, so that we must discuss the case when 

 a, b and c are all even. It is then shown that suitable values can be 

 found for j) and q when the highest power of 2 dividing c is the first 

 or second, and that the case when c is divisible by any power of 2 

 can be reduced to one of the preceding cases. It then follows that 

 H is true for x in the case of a cubic with three real roots. 



Suppose first then that a is odd and that 4i' is the highest power 

 of 4 contained in c. Put c = 4>(7 and take 



q-b = 2V+1 g, p - 2P 

 so that 



CT/4V+1 = C (a + 2P) - Q^ p/4: = q- PK 



Consider first the case when C is even so that it must be divisible 

 by 2 only and not by 4. Then because a is odd, a is not of the form 

 ilf if ^ is even, whatever P may be. 



Also by § (2) we can find a value for P for which p is not of the 

 form M so that H holds for x. 



If, however, C is odd, take P even but Q even or odd according as 



C« = 1 or 3 (mod 4), 



so that a is not of the form M. The only restriction on P is that it 

 should be even, and from § (2) we can find even values of P for 

 which p/i = q — P- is not of the form M when q is given. Hence H 

 holds for ic if a is odd. 



Writing now x = cjy so that H holds for x'li H holds for y and 

 conversely, we see from 



y^ — by^ + acy — c^ = 



that if b is odd, H holds for y and hence for x. Hence we need only 

 consider the case when both a and b are even. 



Should now c be odd, take in (3) p and q both odd, then 



p =3 (mod 8), (7 = 3 (mod 8) 



so that we need only prove H when a, b, c are all even. 



17—2 



