Assignment 10 (pdf)

True/False questions, not to turn in:

• If \(p > 2\) is a prime, and if \(2p+1\) and \(9p+4\) are both prime, then \(\left(\frac{2p+1}{9p+4}\right) = 1\). (Make sure you also have proof one way or the other)
• If \(p > 2\) and \(q = p + 2\) are both prime, then \(\left(\frac{p}{q}\right) = \left(\frac{q}{p}\right)\).
• The product of two "positive residues" is a "positive residue".
• Let \(a\) be relatively prime to the prime \(p\). For each "negative residue" \(k\) compute \(a\cdot k\), and denote by \(g'\) the number of those products that result in "positive residues". Then \(g'=g\).
• Let \(a\) be relatively prime to the prime \(p\). For each "positive residue" \(k\) compute \(a\cdot k\), and denote by \(g'\) the number of those products that result in "positive residues". Then \(g'=g\).

Questions to turn in:

1. Using Euler's identity, we determined the quadratic character for \(-1\), namely the Law of Quadratic Reciprocity for \(\left(\frac{-1}{p}\right)\). Prove the same result by computing \(g\) instead, and using Gauss's Lemma.
2. During the proof that \(T(a,p)=g\bmod 2\), we used that \(a\) is an odd number. What happens when \(a\) is even? Can you demonstrate an example with \(p\) odd, \(a\) even, where \(T(a,p)\neq g\bmod 2\)?
3. Demonstrate that \(T(p, q) + T(q, p) = \frac{p-1}{2}\times \frac{q-1}{2}\) by computing both sides directly, for \(p = 13\) and \(q = 17\).

4. The Law of Quadratic Reciprocity can help us determine if the equation \(x^2+bx+c = 0\) has a solution modulo \(p\). The equation can be rewritten as:

   \[\left(x+\frac{b}{2}\right)^2 = \frac{b^2-4c}{4}\]

   So it will only have a solution if the quantity \(b^2-4c\) is a square modulo \(p\). Use this on the equation \(x^2+13x+4 = 0\) to find: At least two primes \(p\) for which there is a solution and at least two primes \(p\) for which there is not. For the primes for which there is a solution, find it using the above formula. Note that for each prime the equation should have 2 solutions, so you will be finding in total 4 solutions, 2 each for each of the two primes.
5. Compute \(\left(\frac{7}{17}\right)\) in a number of different ways:
   1. Computing all the squares \(\bmod 17\).
   2. Using Euler's Identity and computing \(7^8\bmod 17\).
   3. Using Gauss's Lemma and computing \(g\).
   4. Using Eisenstein's Lemma and computing \(T(7, 17)\).
   5. Using the Laws of Quadratic Reciprocity to reduce the symbol to smaller symbols.

   You should of course get the same result in all cases.