I'm looking for the least odd natural number $k$ such that the equation $x^2\equiv k\pmod {2017}$ doesn't have integer solution.
I'm using the Eisenstein lemma, we have $\big(\frac{a}{p}\big)=(-1)^{T(a,p)}$ when $a$ is an odd integer and $p$ is an odd prime number with $p\nmid a$. Following my calculations we have:
$T(3,2017)=336$ and $T(5,2017)=607$. Since $\big(\frac{3}{1017}\big)=(-1)^{336}=1$ and $\big(\frac{5}{1017}\big)=(-1)^{667}=-1$. So, we have $k=5$.
Am I made any mistakes?
The least quadratic non-residue $\!\!\pmod{p}$ is always a prime. By exaustive search: $$\left(\frac{2}{2017}\right)=1,\quad \left(\frac{3}{2017}\right)=1,\quad \left(\frac{\color{red}{5}}{2017}\right)=-1$$ hence $\color{red}{5}$ is the answer.
$\pm 1,\pm 2,\pm 3$ are surely quadratic residues since $2017$ is a number of the form $24k+1$.
For amusement value: the discriminant $2017$ has class number one. The principal form is $x^2 + 43 xy - 42 y^2,$ equivalent over $SL_2 \mathbb Z$ to $u^2 + uv - 504 v^2.$ Meanwhile, the form does represent $-1$ integrally, which is automatic as $2017 \equiv 1 \pmod 4$ is prime. The Lagrange cycle for the principal form therefore shows all the small primes represented, which are the same as those which are quadratic residues. The line labelled 2 shows that $2$ is represented, line 14 shows that $3$ is represented. Line 40 shows $7.$ Line 32 shows $11.$ The bit about $2$ means that there is an integer solution to $s^2 - 2 t^2 = 2017.$ Look at that, $45^2 - 2 \cdot 2^2 = 2017$
Discriminant $8068$ also has class number one, so $x^2 - 2017 y^2$ does very well. However, it is not possible to have $x^2 - 2017 y^2 \equiv 2 \pmod 4,$ so $2$ itself is out of the picture.
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./indefCycle 1 43 -42
0 form 1 43 -42
1 0
0 1
To Return
1 0
0 1
0 form 1 43 -42 delta -1 ambiguous
1 form -42 41 2 delta 21
2 form 2 43 -21 delta -2
3 form -21 41 4 delta 10
4 form 4 39 -31 delta -1
5 form -31 23 12 delta 2
6 form 12 25 -29 delta -1
7 form -29 33 8 delta 4
8 form 8 31 -33 delta -1
9 form -33 35 6 delta 6
10 form 6 37 -27 delta -1
11 form -27 17 16 delta 1
12 form 16 15 -28 delta -1
13 form -28 41 3 delta 14
14 form 3 43 -14 delta -3
15 form -14 41 6 delta 7
16 form 6 43 -7 delta -6
17 form -7 41 12 delta 3
18 form 12 31 -22 delta -1
19 form -22 13 21 delta 1
20 form 21 29 -14 delta -2
21 form -14 27 23 delta 1
22 form 23 19 -18 delta -1
23 form -18 17 24 delta 1
24 form 24 31 -11 delta -3
25 form -11 35 18 delta 2
26 form 18 37 -9 delta -4
27 form -9 35 22 delta 1
28 form 22 9 -22 delta -1
29 form -22 35 9 delta 4
30 form 9 37 -18 delta -2
31 form -18 35 11 delta 3
32 form 11 31 -24 delta -1
33 form -24 17 18 delta 1
34 form 18 19 -23 delta -1
35 form -23 27 14 delta 2
36 form 14 29 -21 delta -1
37 form -21 13 22 delta 1
38 form 22 31 -12 delta -3
39 form -12 41 7 delta 6
40 form 7 43 -6 delta -7
41 form -6 41 14 delta 3
42 form 14 43 -3 delta -14
43 form -3 41 28 delta 1
44 form 28 15 -16 delta -1
45 form -16 17 27 delta 1
46 form 27 37 -6 delta -6
47 form -6 35 33 delta 1
48 form 33 31 -8 delta -4
49 form -8 33 29 delta 1
50 form 29 25 -12 delta -2
51 form -12 23 31 delta 1
52 form 31 39 -4 delta -10
53 form -4 41 21 delta 2
54 form 21 43 -2 delta -21
55 form -2 41 42 delta 1
56 form 42 43 -1 delta -43
57 form -1 43 42 delta 1 ambiguous -1 composed with form zero
58 form 42 41 -2 delta -21
59 form -2 43 21 delta 2
60 form 21 41 -4 delta -10
61 form -4 39 31 delta 1
62 form 31 23 -12 delta -2
63 form -12 25 29 delta 1
64 form 29 33 -8 delta -4
65 form -8 31 33 delta 1
66 form 33 35 -6 delta -6
67 form -6 37 27 delta 1
68 form 27 17 -16 delta -1
69 form -16 15 28 delta 1
70 form 28 41 -3 delta -14
71 form -3 43 14 delta 3
72 form 14 41 -6 delta -7
73 form -6 43 7 delta 6
74 form 7 41 -12 delta -3
75 form -12 31 22 delta 1
76 form 22 13 -21 delta -1
77 form -21 29 14 delta 2
78 form 14 27 -23 delta -1
79 form -23 19 18 delta 1
80 form 18 17 -24 delta -1
81 form -24 31 11 delta 3
82 form 11 35 -18 delta -2
83 form -18 37 9 delta 4
84 form 9 35 -22 delta -1
85 form -22 9 22 delta 1
86 form 22 35 -9 delta -4
87 form -9 37 18 delta 2
88 form 18 35 -11 delta -3
89 form -11 31 24 delta 1
90 form 24 17 -18 delta -1
91 form -18 19 23 delta 1
92 form 23 27 -14 delta -2
93 form -14 29 21 delta 1
94 form 21 13 -22 delta -1
95 form -22 31 12 delta 3
96 form 12 41 -7 delta -6
97 form -7 43 6 delta 7
98 form 6 41 -14 delta -3
99 form -14 43 3 delta 14
100 form 3 41 -28 delta -1
101 form -28 15 16 delta 1
102 form 16 17 -27 delta -1
103 form -27 37 6 delta 6
104 form 6 35 -33 delta -1
105 form -33 31 8 delta 4
106 form 8 33 -29 delta -1
107 form -29 25 12 delta 2
108 form 12 23 -31 delta -1
109 form -31 39 4 delta 10
110 form 4 41 -21 delta -2
111 form -21 43 2 delta 21
112 form 2 41 -42 delta -1
113 form -42 43 1 delta 43
114 form 1 43 -42
form 1 x^2 + 43 x y -42 y^2
minimum was 1rep x = 1 y = 0 disc 2017 dSqrt 44 M_Ratio 1936
Automorph, written on right of Gram matrix:
-965532677013781482875198088170183835928234466523769 -42440482758478505176147113435578355104819436064102752
-1010487684725678694670169367513770359638558001526256 -44416503120217965353692480891262309300386228532152777
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