Exercise (7.1).1

Find $x^2 \pmod p$ for all the least positive residues of the following. 

In each case plot the graph of $x$ against $x^2 \pmod p$ for:

(a) $p= 5$

(b) $p= 17$

(c) $p= 19$

For each of these primes $p$ solve $x^2 \equiv 2 \pmod p$ and the equivalent quadratic Diophantine equation 

$$x^2 = 2 + py$$

(a) The following table shows the values of $x^2 \pmod 5$.

x	x^2 mod 5
1	1
2	4
3	4
4	1

The following is a plot of $x^2 \pmod 5$ against $x$.

The table and graph show the congruence $x^2 \equiv 2 \pmod 5$ has no solutions, and so the quadratic Diophantine equation $x^2 = 2 + 5y$ has no solutions.

(b) The following table shows the values of $x^2 \pmod {17}$.

x	x^2 mod 17
1	1
2	4
3	9
4	16
5	8
6	2
7	15
8	13
9	13
10	15
11	2
12	8
13	16
14	9
15	4
16	1

The following is a plot of $x^2 \pmod {17}$ against $x$.

The table and graph shows the congruence $x^2 \equiv 2 \pmod 5$ has solutions $x \equiv 6 \pmod {17}$ and $x \equiv 11 \pmod {17}$.

Using $y=\frac{x^2-2}{17}$, the solutions to the Diophantine equation $x^2 = 2 + 17y$ are $(x,y) = (6,2)$ and $(x,y) = (11,7)$.

(c) The following table shows the values of $x^2 \pmod {19}$.

x	x^2 mod 19
1	1
2	4
3	9
4	16
5	6
6	17
7	11
8	7
9	5
10	5
11	7
12	11
13	17
14	6
15	16
16	9
17	4
18	1

The following is a plot of $x^2 \pmod {19}$ against $x$.

The table and graph show the congruence $x^2 \equiv 2 \pmod {19}$ has no solutions, and so the quadratic Diophantine equation $x^2 = 2 + 19y$ has no solutions.