What's the set p,if $37x^2-113y^2=p$ is solvable,with p a prime

Question

if $37x^2-113y^2=p$ is solvable.with p a odd prime. What's the set of all $p$? Does it have a formula?

Answer 1

A book I recommend for this material is Duncan A. Buell, Binary Quadratic Forms and USED COPIES.

This discriminant has class number two. This form is alone in its genus. Also, the principal form represents $-1,$ so there is no distinction between representing $p$ and $-p.$ This form represents $37$ and $113$ and all primes that are nonresidues $\pmod {37}$ and $\pmod {113},$ such as $5,17,19,23,29,\ldots.$ Note that $(2| 113) = 1.$

There are just two classes in this discriminant, and they are in different genera, the form $37 x^2 - 113 y^2$ and the principal form $x^2 - 4181 y^2. $ As $$ 489181478614821790^2 - 4181 \cdot 7565365622404889^2 = -1, $$ any number $n$ is represented by one of these forms if and only if $-n$ is represented by the same form.

Here are the Lagrange cycles. The first coefficient in each triple is a number that is represented primitively. A primitive representation is one with $\gcd(x,y)=1.$ For example, we can solve $x^2 - 4181 y^2 = 4$ with both $x,y$ odd, so that $\gcd(x,y)=1$ in this case, as $$ 985522780227^2 - 4181 \cdot 15241460455^2 = 4. $$ Alright, for each triple $\langle a,b,c \rangle$ in the cycles below, the indicated forms are "reduced." The discriminant is $\Delta = 4 \cdot 37 \cdot 113 = 16724$ so that $\sqrt \Delta \approx 129.32.$ It follows from the definition of reduced that $|a|, b, |c| < 130.$ It is a theorem of Lagrange, Theorem 85 on page 111 of Introduction to the Theory of Numbers by Leonard Eugene Dickson, that any number $n$ that is primitively represented by one of these forms, with $|n| < (1/2) \sqrt \Delta \approx 64.66,$ occurs as the first coefficient of a form in that cycle.

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jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./indefCycle 37 0 -113

  0  form             37           0        -113  delta      0
  1  form           -113           0          37  delta      1
  2  form             37          74         -76


          -1          -1
           0          -1

To Return  
          -1           1
           0          -1

0  form   37 74 -76   delta  -1
1  form   -76 78 35   delta  2
2  form   35 62 -92   delta  -1
3  form   -92 122 5   delta  25
4  form   5 128 -17   delta  -7
5  form   -17 110 68   delta  1
6  form   68 26 -59   delta  -1
7  form   -59 92 35   delta  3
8  form   35 118 -20   delta  -6
9  form   -20 122 23   delta  5
10  form   23 108 -55   delta  -2
11  form   -55 112 19   delta  6
12  form   19 116 -43   delta  -2
13  form   -43 56 79   delta  1
14  form   79 102 -20   delta  -5
15  form   -20 98 89   delta  1
16  form   89 80 -29   delta  -3
17  form   -29 94 68   delta  1
18  form   68 42 -55   delta  -1
19  form   -55 68 55   delta  1
20  form   55 42 -68   delta  -1
21  form   -68 94 29   delta  3
22  form   29 80 -89   delta  -1
23  form   -89 98 20   delta  5
24  form   20 102 -79   delta  -1
25  form   -79 56 43   delta  2
26  form   43 116 -19   delta  -6
27  form   -19 112 55   delta  2
28  form   55 108 -23   delta  -5
29  form   -23 122 20   delta  6
30  form   20 118 -35   delta  -3
31  form   -35 92 59   delta  1
32  form   59 26 -68   delta  -1
33  form   -68 110 17   delta  7
34  form   17 128 -5   delta  -25
35  form   -5 122 92   delta  1
36  form   92 62 -35   delta  -2
37  form   -35 78 76   delta  1
38  form   76 74 -37   delta  -2
39  form   -37 74 76   delta  1
40  form   76 78 -35   delta  -2
41  form   -35 62 92   delta  1
42  form   92 122 -5   delta  -25
43  form   -5 128 17   delta  7
44  form   17 110 -68   delta  -1
45  form   -68 26 59   delta  1
46  form   59 92 -35   delta  -3
47  form   -35 118 20   delta  6
48  form   20 122 -23   delta  -5
49  form   -23 108 55   delta  2
50  form   55 112 -19   delta  -6
51  form   -19 116 43   delta  2
52  form   43 56 -79   delta  -1
53  form   -79 102 20   delta  5
54  form   20 98 -89   delta  -1
55  form   -89 80 29   delta  3
56  form   29 94 -68   delta  -1
57  form   -68 42 55   delta  1
58  form   55 68 -55   delta  -1
59  form   -55 42 68   delta  1
60  form   68 94 -29   delta  -3
61  form   -29 80 89   delta  1
62  form   89 98 -20   delta  -5
63  form   -20 102 79   delta  1
64  form   79 56 -43   delta  -2
65  form   -43 116 19   delta  6
66  form   19 112 -55   delta  -2
67  form   -55 108 23   delta  5
68  form   23 122 -20   delta  -6
69  form   -20 118 35   delta  3
70  form   35 92 -59   delta  -1
71  form   -59 26 68   delta  1
72  form   68 110 -17   delta  -7
73  form   -17 128 5   delta  25
74  form   5 122 -92   delta  -1
75  form   -92 62 35   delta  2
76  form   35 78 -76   delta  -1
77  form   -76 74 37   delta  2
78  form   37 74 -76


  form   37 x^2  + 74 x y  -76 y^2 

minimum was   5rep   x = 3   y = 4 disc   16724 dSqrt 129.32130528  M_Ratio  12.21622
Automorph, written on right of Gram matrix:  
-204735119173764075454348029897491261  -562527184697324302711441753879159120
-273861918865802621056886117020116940  -752458956905369317568120263937725141
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jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ 
    jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ 
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./indefCycle 1 0 -4181

  0  form              1           0       -4181  delta      0
  1  form          -4181           0           1  delta     64
  2  form              1         128         -85


          -1         -64
           0          -1

To Return  
          -1          64
           0          -1

0  form   1 128 -85   delta  -1
1  form   -85 42 44   delta  1
2  form   44 46 -83   delta  -1
3  form   -83 120 7   delta  17
4  form   7 118 -100   delta  -1
5  form   -100 82 25   delta  4
6  form   25 118 -28   delta  -4
7  form   -28 106 49   delta  2
8  form   49 90 -44   delta  -2
9  form   -44 86 53   delta  2
10  form   53 126 -4   delta  -31
11  form   -4 122 115   delta  1
12  form   115 108 -11   delta  -10
13  form   -11 112 95   delta  1
14  form   95 78 -28   delta  -3
15  form   -28 90 77   delta  1
16  form   77 64 -41   delta  -2
17  form   -41 100 41   delta  2
18  form   41 64 -77   delta  -1
19  form   -77 90 28   delta  3
20  form   28 78 -95   delta  -1
21  form   -95 112 11   delta  10
22  form   11 108 -115   delta  -1
23  form   -115 122 4   delta  31
24  form   4 126 -53   delta  -2
25  form   -53 86 44   delta  2
26  form   44 90 -49   delta  -2
27  form   -49 106 28   delta  4
28  form   28 118 -25   delta  -4
29  form   -25 82 100   delta  1
30  form   100 118 -7   delta  -17
31  form   -7 120 83   delta  1
32  form   83 46 -44   delta  -1
33  form   -44 42 85   delta  1
34  form   85 128 -1   delta  -128
35  form   -1 128 85   delta  1
36  form   85 42 -44   delta  -1
37  form   -44 46 83   delta  1
38  form   83 120 -7   delta  -17
39  form   -7 118 100   delta  1
40  form   100 82 -25   delta  -4
41  form   -25 118 28   delta  4
42  form   28 106 -49   delta  -2
43  form   -49 90 44   delta  2
44  form   44 86 -53   delta  -2
45  form   -53 126 4   delta  31
46  form   4 122 -115   delta  -1
47  form   -115 108 11   delta  10
48  form   11 112 -95   delta  -1
49  form   -95 78 28   delta  3
50  form   28 90 -77   delta  -1
51  form   -77 64 41   delta  2
52  form   41 100 -41   delta  -2
53  form   -41 64 77   delta  1
54  form   77 90 -28   delta  -3
55  form   -28 78 95   delta  1
56  form   95 112 -11   delta  -10
57  form   -11 108 115   delta  1
58  form   115 122 -4   delta  -31
59  form   -4 126 53   delta  2
60  form   53 86 -44   delta  -2
61  form   -44 90 49   delta  2
62  form   49 106 -28   delta  -4
63  form   -28 118 25   delta  4
64  form   25 82 -100   delta  -1
65  form   -100 118 7   delta  17
66  form   7 120 -83   delta  -1
67  form   -83 46 44   delta  1
68  form   44 42 -85   delta  -1
69  form   -85 128 1   delta  128
70  form   1 128 -85


  form   1 x^2  + 128 x y  -85 y^2 

minimum was   1rep   x = 1   y = 0 disc   16724 dSqrt 129.32130528  M_Ratio  16724
Automorph, written on right of Gram matrix:  
-4889935136556757385809512072000521  -629142246043060075400954593154322700
-7401673482859530298834759919462620  -952304140942576635636658781763215881
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