NTIC Exercises

Exercises 16.8 Exercises

1.

Fill in all the details of Example 16.0.2 for the congruences \(x^2+5x+5\equiv 0\text{ (mod }5)\) and \(x^2+5x+7\equiv 0\text{ (mod }n)\text{.}\)

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.

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.

Evaluate Legendre symbols for all \(a\neq 0\) where \(p=7\text{,}\) using Euler's Criterion.

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 \(p\text{.}\))