1.
Fill in all the details of Example 16.0.2 for the congruences \(x^2+5x+5\equiv 0\text{ (mod }5)\) and \(x^2+5x+7\equiv 0\text{ (mod }n)\text{.}\)
Exercises 16.8 Exercises
2.
Prove that if \(e>1\text{,}\) then there is no solution to
\begin{equation*}
x^2\equiv -1\text{ (mod }2^e)\text{.}
\end{equation*}
Use our knowledge of squares modulo 4.
3.
For what \(n\) does \(-1\) have a square root modulo \(n\text{?}\) (Hint: use prime factorization and the previous problem along with results earlier in the chapter.)
4.
Clearly \(4\) has a square root modulo \(7\text{.}\) Find all square roots of \(4\) modulo \(7^3\) without using Sage or trying all \(343\) possibilities. Why is this exercise not as challenging as it seems, and what would you do to make it harder?
5.
Solve \(x^2+3x+5\equiv 0\text{ (mod }15)\) using completion of squares and trial and error for square roots.
Exercise Group.
Solve the following congruences without using a computer.
6.
\(x^2+6x+5\equiv 0\) (mod \(17\))
7.
\(5x^2+3x+1\equiv 0\) (mod \(17\))
8.
Prove that if \(p\) is an odd prime
\begin{equation*}
\sum_{a=1}^{p-1}\left(\frac{a}{p}\right)=0\text{.}
\end{equation*}
9.
Explore and conjecture a formula for
\begin{equation*}
\sum_{a\in Q_p} a\text{,}
\end{equation*}
possibly dependent upon some congruence class for \(p\text{.}\)
10.
Show that a quadratic residue can’t be a primitive root if \(p>2\text{.}\)
11.
Show that if \(p\) is an odd prime, then there are exactly \(\frac{p-1}{2}-\phi(p-1)\) residues which are neither QRs nor primitive roots. (Hint: don’t think too hard – just do the obvious counting up.)
12.
Use Euler’s Criterion to find all quadratic residues of 13.
13.
14.
Explore for a pattern for when \(-5\) is a quadratic residue. Try not to use any fancy criteria, but just to seek a pattern based on the number.
15.
Use Euler’s Criterion and the ideas of Proof 16.7.1 to prove that \(3\) has a square root modulo \(p\) if \(p\equiv 1\text{ (mod }12)\text{.}\) (See Proposition 17.3.4 for full details of \(\left(\frac{3}{p}\right)\text{.}\))
16.
Explore for a pattern for, given \(p\text{,}\) how many pairs of consecutive residues are both actually quadratic residues. Then connect this idea to the following formula, which you should evaluate for the same values of \(p\text{:}\)
\begin{equation*}
\sum_{a=1}^{p-2}\left(\frac{a}{p}\right)\left(\frac{a+1}{p}\right)
\end{equation*}
(A harder problem is to prove your evaluation works for all \(p\text{.}\))
17.
Show that, given a power of two, \(2^e\text{,}\) greater than four, \(x^2\equiv a\) (mod \(2^e\)) either has zero or four solutions. (Remark 7.2.7 or even Exercise 7.7.15 may be useful here.)
