hw10 - Math 3320 Problem Set 10 1 exist a form representing...

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Math 3320 Problem Set 10 1 1. Determine the values of Δ for which there exists a quadratic form of discriminant Δ that represents 5, and also determine the discriminants Δ for which there does not exist a form representing 5. 2. Verify that the statement of quadratic reciprocity is true for the following pairs of primes (p,q) : ( 3 , 5 ) , ( 3 , 7 ) , ( 3 , 13 ) , ( 5 , 13 ) , ( 7 , 11 ) , and ( 13 , 17 ) . 3. (a) In the book there is an example near the end of section 2.3 working out which primes are represented by some form of discriminant 13, using quadratic reciprocity for the key step. (This was also done in class.) Do the same thing for discriminant 17. You do not have to repeat the whole argument, just give the steps where the answer is diFerent for 17 than for 13, and give the ±nal answer. (b) Show that all forms of discriminant 17 are equivalent to the principal form x 2 + xy - 4 y 2 . (c) Draw enough of the topograph of
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