HINTS FOR CWORK4: --------- Essay To explain what is a form, you may think of it as a polynomial. Revise the material on the lecture notes: you will find that the essential results are few, and easily stated. Avoid getting into the details of reduction of indefinite forms. --------- Problem 1. (d) Distinguish between the cases: k=1, k=-1 and |k| >1. (f) This is much like problem 3 of cwork2. --------- Problem 3. (a) To be equivalent, forms must have the same discriminant. Definite forms are equivalent to each other if they are equivalent to the unique reduced form in their class. (b) If Q -> Q' via T and Q' -> Q'' via T', then Q -> Q'' via T T' (matrix multiplication). (c) Compute the matrices that transform b -> a and f -> a. Then invert the latter to get a -> f, and multiply it by the former, to get b -> f. --------- Problem 5. (a) Divide into chains by matching the first and last coefficients of the forms. (b) The method of part (a) is ambiguous here: divide into chains using continued fractions. --------- Problem 6. (b) From the representations of 5 and 13, obtain that of 65. From the latter and that of 17, obtain that of 1105, etc. (c) Formula (1) is a special case of (2) with D=1. (d) Make sure you represent 4 and 7, not -4 or -7. --------- Problem 7. (a) If Q(x,y)=x^2+Dy^2=m, then m is the product of two conjugate quadratic integers.