Can I check whether integral solutions exist if I know a rational solution?

Question

The pell-like equation $$x^2-101y^2=-71$$ has the rational solution $(x,y)=(\frac{25}{2},\frac{3}{2})$

Can I use this rational point to find out , whether an integral solution exists ? If yes, can I also find it out for an arbitary equation $$Ax^2+Bxy+Cy^2+Dx+Ey+F=0$$ Again, we can assume that we know a rational solution.

I read somewhere that a rational point allows a parametrization, but I am not sure whether this helps to solve my problem.

Answer 1

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There is a difference between rational solutions and integral solutions, reflected in the form class numbers.

With discriminant $101,$ we have just one class of forms up to $SL_2 \mathbb Z$ equivalence. The Gauss-Lagrange reduced form is $x^2 + 9 xy - 5 y^2.$ The numbers primitively represented are, up to $300,$ these. Note that, since $10^2 - 101 = -1,$ a number $n$ is represented if and only if $-n$ is represented.

Primitively represented positive integers up to  300

           1 =  1 
           5 = 5
          13 = 13
          17 = 17
          19 = 19
          23 = 23
          25 = 5^2
          31 = 31
          37 = 37
          43 = 43
          47 = 47
          65 = 5 * 13
          71 = 71
          79 = 79
          85 = 5 * 17
          95 = 5 * 19
          97 = 97
         101 = 101
         107 = 107
         115 = 5 * 23
         125 = 5^3
         131 = 131
         137 = 137
         155 = 5 * 31
         157 = 157
         169 = 13^2
         179 = 179
         181 = 181
         185 = 5 * 37
         193 = 193
         197 = 197
         211 = 211
         215 = 5 * 43
         221 = 13 * 17
         223 = 223
         227 = 227
         233 = 233
         235 = 5 * 47
         239 = 239
         247 = 13 * 19
         251 = 251
         281 = 281
         283 = 283
         289 = 17^2
         299 = 13 * 23


 Primitively represented positive integers up to  300

           1           9          -5   original form 

========================================================

or discriminant $404,$ there are three classes.

404    factored   2^2 *  101

    1.             1          20          -1   cycle length             2
    2.             4          18          -5   cycle length             6
    3.             5          18          -4   cycle length             6

  form class number is   3

============================================================

The first one, your $x^2 - 101 y^2,$ does not represent $\pm 71$

Primitively represented positive integers up to 300

       1 =  1 
      20 = 2^2 * 5
      37 = 37
      43 = 43
      52 = 2^2 * 13
      65 = 5 * 13
      68 = 2^2 * 17
      76 = 2^2 * 19
      85 = 5 * 17
      92 = 2^2 * 23
      95 = 5 * 19
      97 = 97
     100 = 2^2 * 5^2
     101 = 101
     115 = 5 * 23
     124 = 2^2 * 31
     125 = 5^3
     155 = 5 * 31
     179 = 179
     188 = 2^2 * 47
     221 = 13 * 17
     223 = 223
     233 = 233
     235 = 5 * 47
     247 = 13 * 19
     260 = 2^2 * 5 * 13
     283 = 283
     284 = 2^2 * 71
     299 = 13 * 23

Primitively represented positive integers up to 300

       1          20          -1   original form 

==========================================================

However, $4 x^2 + 18 xy - 5 y^2$ does

Primitively represented positive integers up to  300

           4 = 2^2
           5 = 5
          13 = 13
          17 = 17
          19 = 19
          20 = 2^2 * 5
          23 = 23
          25 = 5^2
          31 = 31
          47 = 47
          52 = 2^2 * 13
          65 = 5 * 13
          68 = 2^2 * 17
          71 = 71
          76 = 2^2 * 19
          79 = 79
          85 = 5 * 17
          92 = 2^2 * 23
          95 = 5 * 19
         100 = 2^2 * 5^2
         107 = 107
         115 = 5 * 23
         124 = 2^2 * 31
         131 = 131
         137 = 137
         148 = 2^2 * 37
         155 = 5 * 31
         157 = 157
         169 = 13^2
         172 = 2^2 * 43
         181 = 181
         185 = 5 * 37
         188 = 2^2 * 47
         193 = 193
         197 = 197
         211 = 211
         215 = 5 * 43
         221 = 13 * 17
         227 = 227
         235 = 5 * 47
         239 = 239
         247 = 13 * 19
         251 = 251
         260 = 2^2 * 5 * 13
         281 = 281
         284 = 2^2 * 71
         289 = 17^2
         299 = 13 * 23


 Primitively represented positive integers up to  300

           4          18          -5   original form 

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