Solutions of Mordell's equation y = x + k

by Helmut Richter



This article contains in several tables all solutions to Mordell's equation y = x + k where 0 < | k | <= 10000 and x <= 1010. All this was done by dumb calculation. There is no theory in this article; hence, there is no proof that these are indeed all solutions to Mordell's equation for these k. When one considers the chance that a given x has a cube with a distance of 10000 or less to a square, it is clear that it is highly improbable that any further solutions exist. In fact, the highest x that yielded a solution (x = 110781386 for k = 8569), is more than an order of magnitude smaller than the range searched. But this is, of course, not a proof of the nonexistence of further solutions. Always when terms like "number of solutions" appears in the sequel, they should be read as "number of found solutions, and very likely, but not certainly, number of all solutions".

Table 1.   Number of known solutions of y = x + k for 0 < k < 1008

Here, for given k, one solution is one value of x so that x + k is a perfect square. About known solutions, see the introduction.

   k    0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

   0       3  1  1  1  1  0  0  4  5  1  0  2  0  0  2  1  8  1  1  0  0  1  0
  24    4  1  1  1  2  0  1  1  0  1  0  1  4  3  1  0  1  1  0  1  2  0  0  0
  48    1  1  1  0  1  0  1  1  1  3  0  0  0  0  0  2  3  4  0  0  2  0  0  1
  72    1  6  0  0  1  0  0  1  4  1  1  0  0  0  0  0  0  4  0  1  1  0  1  0
  96    0  1  1  1  6  2  0  0  0  1  1  1  4  0  0  0  1  6  0  0  0  1  0  1
 120    1  2  1  0  0  1  1  1  2  3  0  1  1  0  1  0  1  0  1  0  0  3  1  1
 144    1  4  0  0  2  0  1  1  1  0  1  0  1  0  0  0  0  4  0  1  3  0  0  0
 168    2  3  1  3  1  0  1  0  0  1  0  0  0  0  0  0  1  1  1  0  1  1  1  0
 192    1  0  0  1  3  2  2  1  0  0  0  0  2  0  1  0  2  0  0  0  0  0  0  0
 216    1  5  0  0  2  0  0  1  1 13  1  0  0  1  0  0  2  4  0  0  1  0  0  0
 240    0  1  0  0  0  0  1  0  2  1  0  1  5  0  0  1  1  1  0  0  4  0  1  0
 264    1  1  0  0  1  2  0  1  1  0  0  1  0  0  0  0  1  3  1  1  0  0  0  0
 288    1  3  1  0  0  0  2  1  1  9  0  0  0  0  0  1  0  0  0  0  0  0  0  0
 312    0  1  0  0  7  0  0  0  1  1  0  1  1  1  0  0  0  1  0  0  1  1  1  0
 336    1  2  0  0  0  0  0  2  1  0  2  1  0  0  2  1  1  4  0  0  1  0  0  3
 360    5  1  1  0  0  0  1  0  1  2  0  0  0  1  0  0  0  4  0  1  1  1  0  0
 384    0  1  1  0  6  1  0  0  3  1  0  0  1  0  0  1  1  1  0  0  2  0  1  1
 408    1  2  0  0  1  0  3  0  1  0  0  0  1  0  0  1  1  1  0  3  0  0  0  0
 432    0  3  0  0  0  0  0  0  1  4  1  2  1  0  0  0  0  4  0  1  0  0  0  0
 456    0  1  0  0  1  0  0  0  3  1  0  0  1  1  0  0  1  0  0  0  1  0  0  0
 480    0  2  0  1  1  2  1  1  0  0  0  0  2  0  0  0  1  0  1  0  1  0  1  0
 504    1  4  1  1  0  0  0  1  5  3  0  0  3  0  1  1  0  4  0  0  0  1  0  0
 528    4  1  1  1  0  0  0  0  0  4  0  1  2  0  0  0  1  1  0  0  5  1  0  1
 552    0  0  0  0  2  1  0  0  0  2  0  1  0  0  1  1  7  0  0  0  0  0  1  2
 576    5  3  0  0  0  0  1  0  1  0  0  0  1  0  0  0  0  3  1  0  0  0  1  1
 600    1  0  1  1  1  1  0  0  0  1  0  0  3  0  0  0  1  1  2  0  1  0  1  0
 624    1  3  1  0  0  0  0  1  1  3  0  1  0  0  1  0  2  0  0  0  1  0  0  0
 648    0  4  0  0  1  0  1  0  2  1  0  2  0  0  0  0  0  3  0  1  1  0  0  0
 672    0  1  0  1  1  1  0  0  0  4  0  0  3  0  1  1  0  1  0  0  0  0  1  0
 696    1  0  0  0  1  2  2  1  1  0  1  0  1  0  1  0  1  1  0  1  2  0  0  1
 720    1  3  0  0  0  0  0  0  1  3  3  0  0  1  0  0  0  4  1  1  2  0  0  1
 744    0  4  1  0  1  0  1  0  0  0  1  0  1  2  0  1  0  0  0  0  1  1  1  0
 768    2  0  0  0  1  0  0  2  1  1  1  0  0  0  0  1  4  2  0  0  0  0  0  0
 792    4  3  0  0  0  0  0  0  0  3  0  0  2  0  1  0  1  0  1  1  1  1  1  0
 816    0  1  0  0  0  0  0  0  1  0  0  1  1  2  0  0  0  1  1  0  5  0  0  0
 840    1  3  1  0  0  0  0  0  1  3  2  0  0  0  1  0  1  2  0  0  0  0  0  1
 864    0  0  0  0  2  0  0  1  1  9  0  0  1  0  1  0  0  0  1  0  0  2  0  0
 888    0  1  1  0  5  0  0  0  0  1  1  3  1  2  0  0  0  1  1  0  1  1  0  0
 912    1  0  0  0  0  0  0  1  1  0  0  0  0  1  0  2  0  0  0  0  1  0  1  0
 936    1  0  0  1  3  0  0  0  0  2  0  0  0  0  0  0  1  3  2  0  0  0  0  0
 960    3  1  1  0  3  0  1  1  1  1  0  1  0  1  0  0  1  1  0  0  0  1  0  0
 984    0  2  0  0  2  0  0  0  0  0  0  0  1  1  0  0  3  3  0  0  2  0  0  0

Table 2.   For 0 < k < 1008: smallest x such that x + k is a square

Note that some of the numbers are not correctly justified lest they touch each other, e.g. the value x = 105 for k = 151.

   k    0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

   0    . -1 -1  1  0 -1  .  . -2 -2 -1  . -2  .  .  1  0 -2  7  5  .  .  3  .
  24   -2  0 -1 -3 -3  . 19 -3  . -2  .  1 -3 -1 11  .  6  2  . -3 -2  .  .  .
  48    1  0 -1  . -3  .  3  9  2 -2  .  .  .  .  . -3 -4 -4  .  . -4  .  .  5
  72   -2 -4  .  . -3  .  . 45 -4  0 -1  .  .  .  .  .  . -4  . -3  2  .  3  .
  96    . 18  7  1 -4 -1  .  .  .  4 15 13 -3  .  .  .  9 -4  .  .  .  3  . 53
 120    1  0 -1  .  . -5 -5 -3 -4 -5  .  5  4  . -5  .  2  . 31  .  . -5  3  1
 144    0 -4  .  . -3  . -5 105 -2 . 23  . 10  .  .  .  . -5  . 33 -4  .  .  .
 168    1  0 -1 -3 14  . -5  .  . -2  .  .  .  .  .  .  6 -4  7  .  2 -5 11  .
 192    4  .  .  1 -3 -1  3  5  .  .  .  . -2  . -5  . -4  .  .  .  .  .  .  .
 216   -6 -6  .  . -6  .  . -3  1 -6 -1  .  .  3  .  . -6 -4  .  .  5  .  .  .
 240    . -6  .  .  .  . -5  .  2 28  . 25 -6  .  .  1  0 -1  .  . -4  .  3  .
 264   -2 -6  .  .  6 -5  . 17  8  .  .  5  .  .  .  . -6  2  7 -3  .  .  .  .
 288    1 -4 -1  .  .  . -5  9 10 -6  .  .  .  .  . 13  .  .  .  .  .  .  .  .
 312    .  6  .  . -6  .  .  . -4 -5  .  1  0 -1  .  .  .  8  .  . -2  7  3  .
 336    4 -6  .  .  .  .  . -7 -7  . 15 -7  .  . -5 -3 -7 -4  .  . 32  .  . -7
 360   -6  0 -1  .  .  . 19  . -7 -2  .  .  .  3  .  .  .  4  . -7 26 -5  .  .
 384    . -6  7  . -4 35  .  . -7 16  .  . 70  .  .  1  0 -1  .  .  5   107 -7
 408   -2  6  .  . -6  . -5  . 17  .  .  .  4  .  . 37 -7 -4  . -3  .  .  .  .
 432    .  2  .  .  .  .  .  .  1 -6 -1 -7 10  .  .  .  . -5  .  5  .  .  .  .
 456    .  3  .  .  6  .  .  . -7  4  .  . -3 15  .  . -6  .  .  .  2  .  .  .
 480    . 12  .  1  0 -1 -5 -7  .  .  .  . -2  .  .  .  9  .  7  .  5  .  3  .
 504   25 -6 47 13  .  .  . -3 -8 -8  .  . -8  . 11 133 . -8  .  .  . -5  .  .
 528   -8  0 -1 85  .  .  .  .  . -8  . -7 -6  .  .  . 33 74  .  . -8  3  .  5
 552    .  .  .  . -3  7  .  .  . -8  . 17  .  . -5  9 -7  .  .  .  .  . 51  1
 576   -8 -6  .  .  .  . 43  . -2  .  .  . 22  .  .  .  . -8 15  .  .  .  3 -7
 600   10  . 23 -3  5 11  .  .  . -5  .  . -8  .  .  . -6  2  7  . 14   219  .
 624    1  0 -1  .  .  .  . 89 -7 -8   269  .  . 67  . -4  .  .  .  8  .  .  .
 648    .  3  .  . -3  . -5  . -8 -6  .  5  .  .  .  .  .  4  . -7  2  .  .  .
 672    . 12  .  1  0 -1  .  .  . -8  .  . -2   179 49  . -4  .  .  .  . 11  .
 696   34  .  .  . -6 -5  3 -3 -7  . 39  . -8  . 19  . 17  8  .  9  5  .  . 13
 720    4  2  .  .  .  .  .  .  1 -9 -9  .  . -9  .  .  . -8 -9 21 -4  .  . -7
 744    . -9  7  . 86  . -5  .  .  . -9  . -3  3  . 25  .  .  .  . 10 -9 27  .
 768   -8  .  .  . 12  .  .  5  2  4 -9  .  .  .  .  1 -7 -1  .  .  .  .  .  .
 792   -6 -9  .  .  .  .  .  .  . -8  .  . 16  . 35  .  6  . -9 -3 149 7  3  .
 816    . 24  .  .  .  .  .  . 38  .  . -7 13 -9  .  .  .  2 31  . -8  .  .  .
 840    1 -6 -1  .  .  .  .  . -4 -2 -9  .  .  . -5  . 14  8  .  .  .  .  . 17
 864    .  .  .  . -3  .  .  9 -7 -9  .  . 37  . 11  .  .  .  7  .  . 19  .  .
 888    . 30 71  . -6  .  .  .  .  4 -9  1  0 -1  .  .  . -4 55  . -2 -5  .  .
 912   -8  .  .  .  .  .  . -7 26  .  .  .  . -9  . -3  .  .  .  .  8  .  3  .
 936   10  .  . 13  6  .  .  .  . -6  .  .  .  .  .  .  9 -8 -9  .  .  .  .  .
 960    1  0 -1  . -4  . -5 69 -7 -2   265  . 11  .  . 12 14  .  .  . 15  .  .
 984    . -9  .  . -3  .  .  .  .  .  .  . -8  3  .   -10 -10 .   -10  .  .  .

Table 3.   Number of known solutions of y = x - k for 0 < k < 1008

Here, for given k, one solution is one value of x so that x - k is a perfect square. About known solutions, see the introduction.

   k    0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

   0       1  1  0  2  0  0  2  1  0  0  2  0  1  0  1  0  0  1  1  1  0  0  1
  24    0  1  2  1  3  0  0  0  0  0  0  1  0  0  0  3  1  0  0  0  1  1  0  3
  48    2  1  0  0  0  2  1  2  1  0  0  0  2  1  0  2  1  0  0  1  0  0  0  1
  72    1  0  1  0  2  0  0  1  0  1  0  1  0  0  0  1  0  1  0  0  0  0  0  1
  96    0  0  0  0  3  0  0  0  3  0  1  1  0  2  0  0  1  0  0  0  4  0  1  0
 120    0  1  0  0  1  1  1  1  1  0  0  0  0  0  0  3  0  0  0  1  0  0  0  1
 144    0  0  1  2  1  0  1  1  3  1  0  1  0  0  0  1  0  0  0  0  0  0  0  1
 168    0  0  1  0  1  0  3  1  0  0  0  0  2  0  0  0  1  0  1  0  1  0  0  3
 192    0  1  0  0  0  0  0  1  3  0  0  0  0  0  0  7  0  0  0  0  2  0  0  2
 216    3  0  0  0  0  0  1  1  0  0  0  0  0  0  0  0  0  1  0  1  1  0  0  1
 240    0  0  2  1  3  1  0  0  0  1  0  1  1  0  0  0  2  0  0  0  0  1  1  0
 264    0  0  0  0  0  0  1  1  0  0  0  0  0  2  0  1  0  0  0  0  1  0  1  1
 288    1  1  0  0  1  1  1  0  0  0  1  1  0  1  0  0  0  0  0  4  1  0  0  0
 312    0  0  0  0  1  0  1  0  0  0  0  0  1  0  0  1  0  1  0  0  0  0  1  0
 336    0  0  0  1  0  0  1  5  0  0  0  0  1  0  1  0  0  0  0  1  1  0  0  1
 360    0  0  2  0  2  0  1  0  7  0  1  1  0  0  0  2  0  0  0  0  0  0  0  0
 384    0  0  0  0  0  0  0  2  0  0  0  0  0  0  0  0  0  0  0  0  0  2  0  0
 408    0  0  0  0  1  0  0  1  0  0  0  0  0  0  0  0  3  1  0  0  0  0  0  9
 432    1  2  0  0  0  0  0  1  3  0  0  0  0  0  0  1  2  0  0  0  0  0  0  1
 456    0  0  0  3  0  0  0  1  1  0  0  0  0  0  0  2  0  1  0  0  2  1  0  0
 480    0  0  0  0  0  0  0  1  0  0  1  0  0  0  0  1  6  0  0  1  0  0  0  8
 504    1  0  1  0  4  0  0  2  1  0  0  1  4  0  0  0  0  0  0  0  1  0  0  0
 528    0  0  0  0  0  1  0  2  0  0  0  0  0  0  0  0  0  0  1  1  0  0  1  0
 552    0  0  0  0  0  1  0  1  1  0  0  0  0  0  1  0  0  0  0  0  1  0  0  0
 576    0  0  0  0  1  0  0  1  0  1  1  1  0  1  0  0  0  2  0  0  1  1  0  1
 600    1  0  1  0  1  0  0  0  1  0  0  0  1  0  0  2  0  0  1  0  0  0  1  0
 624    0  0  0  0  1  1  0  1  0  0  0  0  1  0  0  4  0  0  0  0  0  0  0  1
 648    6  0  0  0  1  0  0  1  0  0  0  0  0  0  0  1  0  1  0  1  0  0  0  1
 672    0  1  2  0  6  0  0  0  2  0  0  1  0  1  0  0  1  0  0  0  0  1  0  0
 696    0  0  0  0  0  0  1  1  3  0  1  0  0  0  0  1  0  1  0  0  0  0  1  1
 720    1  0  0  0  0  1  1  1  2  1  0  0  3  0  0  1  0  0  0  0  0  1  0  0
 744    1  0  0  0  0  0  0  0  0  1  0  3  1  0  0  0  1  0  0  0  1  0  1  2
 768    0  1  0  0  0  0  1  7  1  0  0  0  0  0  2  0  0  0  0  0  0  0  0  1
 792    0  0  0  0  0  1  0  0  1  0  2  0  3  0  0  0  1  0  0  0  0  0  0  0
 816    0  1  0  0  0  0  0  0  0  0  0  0  5  0  0  2  1  0  0  0  0  0  0  0
 840    0  0  0  0  0  0  0  7  0  0  1  0  0  0  0  0  2  0  0  1  1  0  0  0
 864    0  0  1  0  1  0  0  0  0  0  0  2  0  0  0  1  0  0  1  0  0  0  0  1
 888    3  0  1  1  1  0  0  2  0  0  0  0  2  1  0  0  0  0  0  0  1  0  0  0
 912    0  0  2  0  0  0  0  1  0  0  0  0  1  0  0  1  0  0  1  2  1  0  0  1
 936    1  0  0  0  0  0  0  0  5  1  0  0  0  0  0  1  0  0  1  0  0  0  0  0
 960    1  0  0  0  4  0  0  0  1  0  1  1  1  1  1  2  0  0  0  0  5  0  0  0
 984    3  0  0  0  0  0  0  2  0  0  0  0  2  0  0  6  1  0  0  0  0  0  0  1

Table 4.   For 0 < k < 1008: smallest x such that x - k is a square

Note that some of the numbers are not correctly justified lest they touch each other, e.g. the value x = 143 for k = 107. In two places, there was still not enough space: k = 971 with x = 1295 and k = 973 with x = 1297.

   k    0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

   0    .  1  3  .  2  .  .  2  2  .  .  3  . 17  .  4  .  .  3  7  6  .  .  3
  24    .  5  3  3  4  .  .  .  .  .  . 11  .  .  .  4 14  .  .  .  5 21  .  6
  48    4 65  .  .  .  9  7  4 18  .  .  .  4  5  .  4  4  .  . 23  .  .  .  8
  72    6  . 99  .  5  .  . 20  . 13  . 27  .  .  .  7  .  5  .  .  .  .  .  6
  96    .  .  .  .  5  .  .  .  9  . 11 143 .  5  .  .  8  .  .  .  5  .  7  .
 120    .  5  .  .  5  5 15 16 12  .  .  .  .  .  .  6  .  .  . 47  .  .  . 14
 144    .   195  7 197  175  8  6  9  . 51  .  .  . 10  .  .  .  .  .  .  .  6
 168    .  . 59  . 13  .  7 11  .  .  .  .  6  .  .  . 62   163  .  8  .  .  6
 192     257  .  .  .  .  .  7  6  .  .  .  .  .  .  6  .  .  .  .  6  .  .  6
 216    6  .  .  .  .  .  7  8  .  .  .  .  .  .  .  .   377  . 79 21  .  . 15
 240    .  . 11  7 14  9  .  .  . 25  . 83 16  .  .  .  8  .  .  .  . 13  7  .
 264    .  .  .  .  .  . 39 10  .  .  .  .  . 41  .  7  .  .  .  . 12  . 23  8
 288    9 17  .  . 98 57  7  .  .  . 19 399  401  .  .  .  .  .  7 102 .  .  .
 312    .  .  .  .  8  .  7  .  .  .  .  . 10  .  .  7  .  9  .  .  .  .  7  .
 336    .  .  .  7  .  .  7  7  .  .  .  . 13  . 15  .  .  .   119 18  .  . 12
 360    .  . 27  . 29  11815 .  8  . 11 123 .  .  . 10  .  .  .  .  .  .  .  .
 384    .  .  .  .  .  .  .  8  .  .  .  .  .  .  .  .  .  .  .  .  .  9  .  .
 408    .  .  .  .  8  .  . 26  .  .  .  .  .  .  .  . 10 21  .  .  .  .  .  8
 432   12 13  .  .  .  .  . 35  9  .  .  .  .  .  . 22  8  .  .  .  .  .   114
 456    .  .  . 15  .  .  .  8 129 .  .  .  .  .  . 10  .  9  .  .  8 37  .  .
 480    .  .  .  .  .  .  .  8  .  . 11  .  .  .  . 34  8  .   167  .  .  .  8
 504    9   675  .  8  .  .  8  8  .   171 10  .  .  .  .  .  .  . 45  .  .  .
 528    .  .  .  .  .  9  . 14  .  .  .  .  .  .  .  .  .  . 43 11  .  . 31  .
 552    .  .  .  .  . 17  . 10  9  .  .  .  .  . 15  .  .  .  .  . 12  .  .  .
 576    .  .  .   194  .  . 38  . 9 115 783  785  .  .  . 33  .   198 13  . 24
 600   10  . 11  . 20  .  .  .  9  .  .  . 21  .  . 16  .  . 19  .  .   127  .
 624    .  .  .  . 14  9  . 58  .  .  .  . 28  .  . 10  .  .  .  .  .  .  . 42
 648    9  .  .  . 53  .  . 11  .  .  .  .  .  .  . 94  .  9   223  .  .  . 15
 672    . 29 75  . 10  .  .  .  9  .   227  . 49  .  . 17  .  .  .  .  9  .  .
 696    .  .  .  .  .   103 112 9  . 11  .  .  .  . 10  .  9  .  .  .  . 23 14
 720    9  .  .  .  .  9 55 32  9  9  .  . 16  .   106  .  .  .  .  . 25  .  .
 744   10  .  .  .  .  .  .  .  . 13  . 11 30  .  .   254  .  .  . 92  1183 12
 768     1025 .  .  .  . 15 10 258 .  .  .  .  . 47  .  .  .  .  .  .  .  . 18
 792    .  .  .  .  . 21  .  . 41  . 11  . 10  .  .  . 14  .  .  .  .  .  .  .
 816    . 17  .  .  .  .  .  .  .  .  .  . 12  .  . 10 68  .  .  .  .  .  .  .
 840    .  .  .  .  .  .  . 11  .  . 35  .  .  .  .  . 10  .   287 36  .  .  .
 864    .   1155  1157 .  .  .  .  .  . 15  .  .  . 10  .  . 51  .  .  .  . 12
 888   34  . 11 31 56  .  . 14  .  .  .  . 10 13  .  .  .  .  .  . 77  .  .  .
 912    .  . 27  .  .  .  . 10  .  .  .  . 37  .  . 28  .  . 19 11 18  .  . 26
 936   10  .  .  .  .  .  .  . 12 141 .  .  .  .  . 10  .   187  .  .  .  .  .
 960   16  .  .  . 10  .  .  . 33  . 11 ** 13 ** 15 10  .  .  .  . 14  .  .  .
 984   10  .  .  .  .  .  . 10  .  .  .  . 10  .  . 10 10  .  .  .  .  .  . 11

Table 5.   Solutions of y = x + k for 0 < | k | < 1008

This table contains the solutions for 0 < | k | < 1008 and x <= 1010 for those k for which there is more than one x satifying the equation. The values of x for those k having a unique corresponding x can be found in table 2 for k > 0 and in table 4 for k < 0.

    k #sol (x,y)

    1   3  (-1,0), (0,1), (2,3)
   -4   2  (2,2), (5,11)
   -7   2  (2,1), (32,181)
    8   4  (-2,0), (1,3), (2,4), (46,312)
    9   5  (-2,1), (0,3), (3,6), (6,15), (40,253)
  -11   2  (3,4), (15,58)
   12   2  (-2,2), (13,47)
   15   2  (1,4), (109,1138)
   17   8  (-2,3), (-1,4), (2,5), (4,9), (8,23), (43,282), (52,375),
	   (5234,378661)
   24   4  (-2,4), (1,5), (10,32), (8158,736844)
  -26   2  (3,1), (35,207)
   28   2  (-3,1), (2,6)
  -28   3  (4,6), (8,22), (37,225)
   36   4  (-3,3), (0,6), (4,10), (12,42)
   37   3  (-1,6), (3,8), (243,3788)
  -39   3  (4,5), (10,31), (22,103)
   44   2  (-2,6), (5,13)
  -47   3  (6,13), (12,41), (63,500)
  -48   2  (4,4), (28,148)
  -53   2  (9,26), (29,156)
  -55   2  (4,3), (56,419)
   57   3  (-2,7), (4,11), (7,20)
  -60   2  (4,2), (136,1586)
   63   2  (-3,6), (1,8)
  -63   2  (4,1), (568,13537)
   64   3  (-4,0), (0,8), (8,24)
   65   4  (-4,1), (-1,8), (14,53), (584,14113)
   68   2  (-4,2), (152,1874)
   73   6  (-4,3), (2,9), (3,10), (6,17), (72,611), (356,6717)
  -76   2  (5,7), (101,1015)
   80   4  (-4,4), (1,9), (4,12), (44,292)
   89   4  (-4,5), (-2,9), (10,33), (55,408)
  100   6  (-4,6), (0,10), (5,15), (20,90), (24,118), (2660,137190)
 -100   3  (5,5), (10,30), (34,198)
  101   2  (-1,10), (95,926)
 -104   3  (9,25), (30,164), (42,272)
  108   4  (-3,9), (-2,10), (6,18), (366,7002)
 -109   2  (5,4), (145,1746)
  113   6  (-4,7), (2,11), (8,25), (11,38), (26,133), (422,8669)
 -116   4  (5,3), (6,10), (38,234), (158,1986)
  121   2  (0,11), (12,43)
  128   2  (-4,8), (17,71)
  129   3  (-5,2), (-2,11), (16,65)
 -135   3  (6,9), (19,82), (24,117)
  141   3  (-5,4), (7,22), (3067,169852)
  145   4  (-4,9), (-1,12), (6,19), (54,397)
 -147   2  (7,14), (91,868)
  148   2  (-3,11), (21,97)
 -152   3  (6,8), (17,69), (26,132)
  161   4  (-5,6), (2,13), (4,15), (190,2619)
  164   3  (-4,10), (5,17), (8,26)
  168   2  (1,13), (22,104)
  169   3  (0,13), (3,14), (78,689)
  171   3  (-3,12), (9,30), (937,28682)
 -174   3  (7,13), (799,22585), (5215,376601)
 -180   2  (6,6), (69,573)
 -191   3  (6,5), (255,4072), (810,23053)
  196   3  (-3,13), (0,14), (84,770)
  197   2  (-1,14), (19,84)
  198   2  (3,15), (27,141)
 -200   3  (6,4), (9,23), (66,536)
  204   2  (-2,14), (13,49)
 -207   7  (6,3), (12,39), (18,75), (31,172), (312,5511), (331,6022),
	   (367806,223063347)
  208   2  (-4,12), (12,44)
 -212   2  (6,2), (717,19199)
 -215   2  (6,1), (2904,156493)
 -216   3  (6,0), (10,28), (33,189)
  217   5  (-6,1), (2,15), (8,27), (39,244), (2928,158437)
  220   2  (-6,2), (741,20171)
  225  13  (-6,3), (-5,10), (0,15), (4,17), (6,21), (10,35), (15,60),
	   (30,165), (60,465), (180,2415), (336,6159), (351,6576),
	   (720114,611085363)
  232   2  (-6,4), (9,31)
  233   4  (-4,13), (-2,15), (7,24), (202,2871)
 -242   2  (11,33), (323,5805)
 -244   3  (14,50), (22,102), (325,5859)
  248   2  (2,16), (41,263)
  252   5  (-6,6), (-3,15), (18,78), (58,442), (93,897)
 -256   2  (8,16), (20,88)
  260   4  (-4,14), (4,18), (16,66), (29,157)
  269   2  (-5,12), (11,40)
 -277   2  (41,262), (317,5644)
  281   3  (2,17), (14,55), (20,91)
  289   3  (-4,15), (0,17), (68,561)
  294   2  (-5,13), (211,3065)
  297   9  (-6,9), (-2,17), (3,18), (4,19), (12,45), (34,199), (48,333),
	   (1362,50265), (93844,28748141)
 -307   4  (7,6), (11,32), (71,598), (939787,911054064)
  316   7  (-6,10), (-3,17), (2,18), (5,21), (50,354), (90,854), (162,2062)
  337   2  (-6,11), (24,119)
  343   2  (-7,0), (21,98)
 -343   5  (7,0), (8,13), (14,49), (28,147), (154,1911)
  346   2  (15,61), (159,2005)
  350   2  (-5,15), (11,41)
  353   4  (-4,17), (2,19), (38,235), (117188,40116655)
  359   3  (-7,4), (5,22), (73,624)
  360   5  (-6,12), (1,19), (6,24), (9,33), (346,6436)
 -362   2  (27,139), (483,10615)
 -364   2  (29,155), (485,10681)
 -368   7  (8,12), (9,19), (24,116), (32,180), (48,332), (944,29004),
	   (1313,47577)
  369   2  (-2,19), (10,37)
 -375   2  (10,25), (16,61)
  377   4  (4,21), (22,105), (23,112), (47044,10203669)
  388   6  (-4,18), (-3,19), (8,30), (12,46), (341,6297), (1376,51042)
 -391   2  (8,11), (50,353)
  392   3  (-7,7), (2,20), (14,56)
  404   2  (5,23), (13,51)
 -405   2  (9,18), (61,476)
  409   2  (6,25), (18,79)
  414   3  (-5,17), (3,21), (3075,170517)
 -424   3  (10,24), (17,67), (142,1692)
  427   3  (-3,20), (9,34), (30333,5282908)
 -431   9  (8,9), (11,30), (20,87), (30,163), (36,215), (138,1621),
	   (150,1837), (575,13788), (3903,243836)
  433   3  (2,21), (11,42), (36,217)
 -433   2  (13,42), (577,13860)
 -440   3  (9,17), (14,48), (146,1764)
  441   4  (-6,15), (0,21), (7,28), (42,273)
  443   2  (-7,10), (77,676)
 -448   2  (8,8), (128,1448)
  449   4  (-5,18), (-2,21), (8,31), (176,2335)
 -459   3  (15,54), (19,80), (67,548)
  464   3  (-7,11), (-4,20), (20,92)
 -471   2  (10,23), (4528,304691)
 -476   2  (8,6), (240,3718)
  481   2  (12,47), (27,142)
  485   2  (-1,22), (31,174)
  492   2  (-2,22), (118,1282)
 -496   6  (8,4), (16,60), (25,123), (40,252), (113,1201), (560,13252)
 -503   8  (8,3), (12,35), (18,73), (23,108), (44,291), (134,1551),
	   (294,5041), (1008,32003)
  505   4  (-6,17), (-4,21), (14,57), (371,7146)
 -508   4  (8,2), (284,4786), (677,17615), (2288,109442)
 -511   2  (8,1), (9200,882433)
  512   5  (-8,0), (-7,13), (4,24), (8,32), (184,2496)
  513   3  (-8,1), (6,27), (9232,887041)
  516   3  (-8,2), (40,254), (2320,111746)
 -516   4  (10,22), (13,41), (181,2435), (418,8546)
  521   4  (-8,3), (2,23), (10,39), (1040,33539)
  528   4  (-8,4), (1,23), (16,68), (592,14404)
 -535   2  (14,47), (2156,100109)
  537   4  (-8,5), (-2,23), (19,86), (124,1381)
  540   2  (-6,18), (21,99)
  548   5  (-8,6), (-4,22), (28,150), (61,477), (272,4486)
  556   2  (-3,23), (30,166)
  561   2  (-8,7), (4,25)
  568   7  (-7,15), (2,24), (6,28), (18,80), (57,431), (161,2043),
	   (1137,38339)
  575   2  (1,24), (29,158)
  576   5  (-8,8), (0,24), (12,48), (24,120), (160,2024)
  577   3  (-6,19), (-1,24), (8,33)
  593   3  (-8,9), (-4,23), (76,663)
 -593   2  (33,188), (909,27406)
  612   3  (-8,10), (4,26), (13,53)
 -615   2  (16,59), (46,311)
  618   2  (7,31), (421351,273505487)
  625   3  (0,25), (6,29), (75,650)
  633   3  (-8,11), (-2,25), (46,313)
 -639   4  (10,19), (12,33), (27,138), (654,16725)
  640   2  (-4,24), (9,37)
 -648   6  (9,9), (18,72), (22,100), (54,396), (97,955), (1809,76941)
  649   4  (3,26), (20,93), (26,135), (1398,52271)
  656   2  (-8,12), (80,716)
  659   2  (5,28), (5393,396046)
  665   3  (4,27), (16,69), (44,293)
 -674   2  (75,649), (899,26955)
 -676   6  (10,18), (13,39), (26,130), (130,1482), (338,6214), (901,27045)
 -680   2  (9,7), (8394,769048)
  681   4  (-8,13), (7,32), (10,41), (82,743)
  684   3  (-2,26), (6,30), (45,303)
  701   2  (-5,24), (247,3882)
  702   2  (3,27), (139,1639)
 -704   3  (9,5), (12,32), (60,464)
  716   2  (5,29), (110,1154)
  721   3  (2,27), (15,64), (32,183)
 -728   2  (9,1), (74,636)
  729   3  (-9,0), (0,27), (18,81)
  730   3  (-9,1), (-1,27), (231,3511)
 -732   3  (16,58), (52,374), (76,662)
  737   4  (-8,15), (-2,27), (14,59), (59,454)
  740   2  (-4,26), (3296,189226)
  745   4  (-9,4), (-6,23), (6,31), (96,941)
 -755   3  (11,24), (39,242), (891,26596)
  757   2  (3,28), (2063,93702)
 -767   2  (12,31), (1023,32720)
  768   2  (-8,16), (52,376)
  775   2  (5,30), (41,264)
 -775   7  (10,15), (19,78), (20,85), (70,585), (80,715), (16750,2167815),
	   (26530,4321215)
 -782   2  (47,321), (87,811)
  784   4  (-7,21), (0,28), (8,36), (56,420)
  785   2  (-1,28), (11,46)
  792   4  (-6,24), (-2,28), (9,39), (177,2355)
  793   3  (-9,8), (-4,27), (62,489)
  801   3  (-8,17), (-5,26), (22,107)
 -802   2  (11,23), (307,5379)
  804   2  (16,70), (88,826)
 -804   3  (10,14), (82,742), (157,1967)
 -828   5  (12,30), (13,37), (24,114), (108,1122), (4464,298254)
  829   2  (-9,10), (23,114)
 -831   2  (10,13), (220,3263)
  836   5  (-8,18), (4,30), (5,31), (20,94), (3712,226158)
  841   3  (-6,25), (0,29), (87,812)
 -847   7  (11,22), (16,57), (22,99), (86,797), (88,825), (638,16115),
	   (657547,533200074)
  849   3  (-2,29), (10,43), (28,151)
  850   2  (-9,11), (15,65)
 -856   2  (10,12), (230,3488)
  857   2  (8,37), (104,1061)
  868   2  (-3,29), (36,218)
  873   9  (-9,12), (-8,19), (3,30), (6,33), (12,51), (66,537), (178,2375),
	   (432,8979), (978,30585)
 -875   2  (15,50), (291,4964)
  885   2  (19,88), (68239,17825798)
 -888   3  (34,196), (73,623), (334,6104)
  892   5  (-6,26), (2,30), (18,82), (29,159), (53,387)
 -895   2  (14,43), (116,1249)
  899   3  (1,30), (5,32), (11393,1216066)
 -900   2  (10,10), (205,2935)
  901   2  (-1,30), (467,10092)
 -914   2  (27,137), (1779,75035)
  927   2  (-3,30), (33,192)
 -931   2  (11,20), (23,106)
  940   3  (6,34), (21,101), (54,398)
 -944   5  (12,28), (17,63), (20,84), (164,2100), (2364,114940)
  945   2  (-6,27), (16,71)
  953   3  (-8,21), (2,31), (7,36)
  954   2  (-9,15), (63,501)
  960   3  (1,31), (4,32), (436,9104)
  964   3  (-4,30), (5,33), (48,334)
 -964   4  (10,6), (322,5778), (605,14881), (4402,292062)
 -975   2  (10,5), (880,26105)
 -980   5  (14,42), (21,91), (29,153), (126,1414), (326,5886)
 -984   3  (10,4), (25,121), (769,21325)
  985   2  (-9,16), (1011,32146)
  988   2  (-3,31), (42,274)
 -991   2  (10,3), (2480,123503)
 -996   2  (10,2), (5605,419627)
 -999   6  (10,1), (12,27), (40,251), (147,1782), (174,2295), (22480,3370501)
 1000   3  (-10,0), (-6,28), (65,525)
 1001   3  (-10,1), (92,883), (22520,3379501)
 1004   2  (-10,2), (5645,424127)

Table 6.   Solutions of y = x + k for 1008 <= | k | <= 10000

This table contains the solutions for 1008 <= | k | <= 10000 and x <= 1010 for those k for which there are at least four x satifying the equation. The values of x for those k having less corresponding x can only be found via the big table.

    k #sol (x,y)

 1009   5  (-10,3), (6,35), (8,39), (1355,49878), (2520,126503)
 1016   7  (-10,4), (2,32), (17,77), (22,108), (25,129), (1330,48504),
	   (6194,487480)
 1025  16  (-10,5), (-5,30), (-4,31), (-1,32), (4,33), (10,45), (20,95),
	   (40,255), (50,355), (64,513), (155,1930), (166,2139), (446,9419),
	   (920,27905), (3631,218796), (3730,227805)
-1071   6  (15,48), (16,55), (18,69), (1488,57399), (3810,235173),
	   (10578,1087941)
 1088   9  (-8,24), (-4,32), (1,33), (8,40), (16,72), (32,184), (172,2256),
	   (208,3000), (20936,3029288)
 1100   4  (-10,10), (5,35), (14,62), (245,3835)
 1116   4  (-6,30), (-3,33), (10,46), (450,9546)
-1192   4  (17,61), (26,128), (398,7940), (153761,60293333)
-1208   4  (18,68), (78,688), (249,3929), (402,8060)
 1224   4  (1,35), (18,84), (30,168), (393,7791)
 1225   5  (-10,15), (0,35), (14,63), (35,210), (120,1315)
 1296   4  (-8,28), (0,36), (9,45), (72,612)
 1304   6  (-7,31), (-2,36), (10,48), (41,265), (350,6548), (2665,137577)
 1305   9  (-9,24), (-6,33), (4,37), (6,39), (24,123), (36,219), (51,366),
	   (376,7291), (1434,54303)
 1412   4  (-11,9), (-8,30), (68,562), (188,2578)
-1439   8  (12,17), (15,44), (20,81), (32,177), (54,395), (122,1347),
	   (590,14331), (445650,297502669)
 1513   4  (-9,28), (2,39), (8,45), (186,2537)
 1536   5  (-8,32), (4,40), (25,131), (40,256), (32632,5894752)
 1548   4  (-3,39), (6,42), (78,690), (493278,346447650)
 1585   4  (-6,37), (-4,39), (11,54), (1454,55443)
-1588   4  (14,34), (29,151), (2117,97405), (2933,158843)
-1607   4  (12,11), (18,65), (51,362), (3642,219791)
-1664   4  (12,8), (17,57), (140,1656), (705,18719)
-1692   4  (12,6), (21,87), (48,330), (1272,45366)
-1712   6  (12,4), (33,185), (36,212), (132,1516), (156,1948), (2892,155524)
-1719   4  (12,3), (30,159), (39,240), (5160,370659)
-1724   5  (12,2), (24,110), (45,299), (1749,73145), (11640,1255826)
-1727   7  (12,1), (27,134), (42,269), (56,417), (278,4635), (2303,110520),
	   (46632,10069921)
 1729   5  (-12,1), (-10,27), (191,2640), (218,3219), (46680,10085473)
 1737  11  (-12,3), (-8,35), (-6,39), (3,42), (18,87), (54,399), (67,550),
	   (84,771), (1383,51432), (5208,375843), (572034,432646071)
 1753   4  (-12,5), (-9,32), (12,59), (102,1031)
 1764   5  (-12,6), (0,42), (21,105), (28,154), (1320,47958)
 1772   5  (-11,21), (-2,42), (13,63), (38,238), (62,490)
 1809   6  (-12,9), (6,45), (10,53), (15,72), (148,1801), (600,14697)
 1872   7  (-12,12), (4,44), (9,51), (12,60), (153,1893), (348,6492),
	   (1348,49492)
 1897   4  (-12,13), (-6,41), (99,986), (228,3443)
 1900   6  (-10,30), (5,45), (6,46), (30,170), (45,305), (8270,752070)
 1961   4  (-10,31), (4,45), (7,48), (950,29281)
-1999   6  (22,93), (40,249), (74,635), (100,999), (299,5170), (562,13323)
 2024   5  (-10,32), (-7,41), (1,45), (26,140), (58,444)
 2025   4  (-9,36), (0,45), (10,55), (90,855)
 2033   4  (-8,39), (-2,45), (11,58), (206,2957)
-2036   4  (21,85), (30,158), (678,17654), (918,27814)
 2052   6  (-12,18), (-3,45), (4,46), (24,126), (64,514), (168,2178)
-2071   6  (16,45), (20,77), (28,141), (35,202), (118,1281), (2338,113049)
 2089  14  (-12,19), (-10,33), (-4,45), (3,46), (8,51), (18,89), (60,467),
	   (71,600), (80,717), (170,2217), (183,2476), (698,18441),
	   (9278,893679), (129968,46854861)
 2185   4  (6,49), (36,221), (39,248), (156,1949)
-2188   5  (13,3), (29,149), (53,383), (2917,157545), (22549,3386031)
 2201   5  (-13,2), (2,47), (20,101), (32,187), (524,11995)
 2241   6  (-6,45), (-5,46), (12,63), (30,171), (127,1432), (8292,755073)
 2250   4  (-9,39), (15,75), (31,179), (10119,1017903)
 2296   5  (-10,36), (2,48), (9,55), (57,433), (534,12340)
 2304   5  (-12,24), (0,48), (16,80), (48,336), (105,1077)
 2305   4  (-1,48), (24,127), (26,141), (17906,2396061)
 2312   5  (-2,48), (17,85), (34,204), (89,841), (238,3672)
-2351  10  (15,32), (18,59), (30,157), (48,329), (276,4585), (378,7349),
	   (558,13181), (651,16610), (3135,175532), (13035,1488218)
 2368   5  (-7,45), (-4,48), (12,64), (41,267), (972,30304)
 2404   4  (-12,26), (8,54), (20,102), (141,1675)
 2521   4  (-13,18), (-10,39), (72,613), (384,7525)
 2537   4  (-8,45), (4,51), (68,563), (166864,68162259)
-2548   4  (14,14), (22,90), (133,1533), (413,8393)
-2575   4  (14,13), (16,39), (139,1638), (646,16419)
 2600   5  (-10,40), (1,51), (10,60), (25,135), (134,1552)
 2628   9  (-12,30), (-8,46), (-3,51), (12,66), (16,82), (36,222), (96,942),
	   (381,7437), (1341,49107)
-2708   4  (14,6), (902,27090), (2373,115597), (6134,480414)
 2745   4  (-14,1), (4,53), (19,98), (86464,25424533)
 2793   4  (-14,7), (7,56), (16,83), (1792,75859)
 2817  10  (-12,33), (-6,51), (-2,53), (18,93), (24,129), (27,150),
	   (214,3131), (684,17889), (1327,48340), (8598,797253)
 2852   4  (-11,39), (4,54), (8,58), (772,21450)
-2888   4  (17,45), (38,228), (114,1216), (209,3021)
 2913   4  (-14,13), (-8,49), (58,445), (286,4837)
 2969   4  (-14,15), (8,59), (10,63), (3703,225336)
 3012   4  (-11,41), (-8,50), (28,158), (292,4990)
 3025  11  (-10,45), (-6,53), (0,55), (11,66), (15,80), (20,105), (44,297),
	   (66,539), (110,1155), (330,5995), (1334,48723)
 3033   8  (-14,17), (-9,48), (-2,55), (6,57), (12,69), (52,379), (192,2661),
	   (423,8700)
-3051   5  (15,18), (51,360), (115,1232), (303,5274), (353103,209822526)
 3097   4  (-13,30), (-12,37), (74,639), (4514,303279)
 3132   4  (-6,54), (13,73), (34,206), (14493,1744767)
 3185   5  (-14,21), (4,57), (14,77), (224,3353), (7114,600027)
 3209   4  (-10,47), (8,61), (23,124), (38,241)
-3231   4  (15,12), (18,51), (136,1585), (408,8241)
-3332   6  (18,50), (21,77), (42,266), (93,895), (882,26194), (10437,1066261)
 3356   6  (-11,45), (2,58), (5,59), (10,66), (610,15066), (1514,58910)
-3376   4  (20,68), (73,621), (76,660), (182180,77759068)
 3384   4  (-15,3), (6,60), (18,96), (145,1747)
 3428   4  (-8,54), (-4,58), (13,75), (796,22458)
-3471   5  (16,25), (160,2023), (235,3602), (5230,378227), (43510,9075773)
 3489   5  (-5,58), (-2,59), (10,67), (1528,59729), (1978,87971)
 3592   5  (-7,57), (2,60), (98,972), (174,2296), (1577,62625)
 3600   4  (-15,15), (0,60), (24,132), (40,260)
-3623   4  (18,47), (24,101), (39,236), (2334,112759)
 3648   4  (-8,56), (16,88), (28,160), (4153,267635)
 3664   8  (-15,17), (-12,44), (-4,60), (20,108), (33,199), (68,564),
	   (108,1124), (185,2517)
 3713   4  (2,61), (8,65), (32,191), (431,8948)
 3736   4  (-15,19), (17,93), (62,492), (90,856)
 3753   6  (-12,45), (6,63), (7,64), (42,279), (16116,2045907),
	   (5024238,11261735055)
-3807  11  (16,17), (18,45), (27,126), (36,207), (126,1413), (142,1691),
	   (162,2061), (316,5617), (927,28224), (1306,47197), (37368,7223535)
 3844   8  (-12,46), (0,62), (5,63), (8,66), (93,899), (248,3906),
	   (620,15438), (1836,78670)
-3896  11  (18,44), (30,152), (33,179), (41,255), (50,348), (110,1152),
	   (465,10027), (545,12723), (878,26016), (1298,46764), (4398,291664)
-3952   5  (16,12), (17,31), (328,5940), (992,31244), (1816,77388)
-3967   5  (47,316), (146,1763), (248,3905), (332,6049), (1286,46117)
 3969   9  (-14,35), (-5,62), (0,63), (18,99), (28,161), (36,225), (63,504),
	   (270,4437), (630,15813)
-4031   5  (20,63), (63,496), (86,795), (120,1313), (3638,219429)
 4032   4  (-12,48), (4,64), (9,69), (57,435)
 4033   4  (-4,63), (26,147), (27,154), (90548,27246975)
-4080   5  (16,4), (49,337), (64,508), (1096,36284), (9184,880132)
-4087   7  (16,3), (22,81), (131,1498), (158,1985), (308,5405), (1082,35591),
	   (16352,2091011)
 4100   4  (-16,2), (5,65), (20,110), (36896,7087106)
 4112  10  (-16,4), (-8,60), (8,68), (17,95), (64,516), (73,627), (88,828),
	   (2984,163004), (9248,889348), (15992,2022340)
 4160   4  (-16,8), (-4,64), (56,424), (2336,112904)
 4217   4  (-16,11), (2,65), (23,128), (44,299)
-4220   6  (21,71), (24,98), (36,206), (56,414), (176,2334), (869,25617)
 4265   4  (-16,13), (-14,39), (199,2808), (706,18759)
 4312  10  (-7,63), (-6,64), (9,71), (14,84), (42,280), (78,692), (113,1203),
	   (609,15029), (938,28728), (16142,2050860)
 4329   7  (-12,51), (-9,60), (3,66), (10,73), (30,177), (48,339),
	   (1390,51823)
 4356   8  (-11,55), (-8,62), (0,66), (12,78), (45,309), (132,1518),
	   (264,4290), (1540,60434)
 4364   5  (-10,58), (-2,66), (5,67), (13,81), (358,6774)
 4420   4  (-16,18), (-4,66), (36,226), (69,577)
 4481  12  (-10,59), (-8,63), (-5,66), (2,67), (14,85), (22,123), (32,193),
	   (175,2316), (364,6945), (640,16191), (1862,80347), (3739,228630)
-4598   4  (23,87), (47,315), (543,12653), (1479,56879)
 4600   5  (-15,35), (-10,60), (9,73), (50,360), (386,7584)
 4625   5  (-16,23), (-1,68), (10,75), (26,149), (160,2025)
 4672   7  (-16,24), (8,72), (12,80), (24,136), (137,1605), (288,4888),
	   (1424,53736)
 4721   4  (-16,25), (-10,61), (62,493), (290,4939)
-4743   4  (18,33), (19,46), (132,1515), (6204,488661)
-4799   7  (24,95), (27,122), (30,149), (72,607), (90,851), (222,3307),
	   (6399,511880)
 4825   7  (-16,27), (-9,64), (-4,69), (6,71), (14,87), (84,773), (194,2703)
 4964   5  (-4,70), (8,74), (140,1658), (27320,4515658), (1588448,2001978934)
 4977   8  (-17,8), (-12,57), (-6,69), (4,71), (22,125), (72,615), (198,2787),
	   (459,9834)
 5057   4  (-17,12), (-16,31), (394,7821), (1882,81645)
-5095   5  (19,42), (44,283), (64,507), (106,1089), (1274,45473)
 5113   4  (-13,54), (6,73), (8,75), (5462,403671)
 5120   4  (-16,32), (4,72), (16,96), (176,2336)
-5296   5  (20,52), (68,556), (113,1199), (1340,49052), (10916,1140500)
-5319   4  (22,73), (60,459), (114,1215), (787,22078)
 5328   6  (-12,60), (1,73), (12,84), (36,228), (121,1333),
	   (233796,113046108)
 5400   8  (-15,45), (-6,72), (10,80), (25,145), (30,180), (129,1467),
	   (190,2620), (4170,269280)
 5412   4  (-8,70), (4,74), (148,1802), (1775104,2365024826)
-5508   4  (18,18), (202,2870), (693,18243), (954,29466)
-5543   6  (18,17), (24,91), (32,165), (119,1296), (282,4735), (968,30117)
 5561   4  (-13,58), (4,75), (10,81), (662,17033)
 5624   5  (-10,68), (1,75), (118,1284), (178,2376), (3425,200443)
 5696   5  (-16,40), (-8,72), (17,103), (40,264), (220,3264)
 5776   5  (-15,49), (0,76), (24,140), (57,437), (1824,77900)
-5823   4  (18,3), (28,127), (123,1362), (26208,4242783)
 5832   4  (-18,0), (9,81), (18,108), (414,8424)
 5841  10  (-18,3), (-8,73), (-6,75), (12,87), (15,96), (60,471), (75,654),
	   (3694,224515), (5490,406779), (26280,4260279)
 5868   4  (-18,6), (6,78), (21,123), (6597,535821)
-5872   4  (32,164), (56,412), (296,5092), (488,10780)
-5887   4  (22,69), (58,435), (116,1247), (667,17226)
 5913   5  (-18,9), (-9,72), (76,667), (108,1125), (2952,160389)
 5937   4  (-17,32), (-2,77), (28,167), (724,19481)
-5975   5  (20,45), (30,145), (50,345), (99,982), (311,5484)
 5976   4  (-18,12), (-15,51), (202,2872), (474,10320)
-6012   4  (21,57), (72,606), (76,658), (99708,31484370)
 6057   5  (-18,15), (3,78), (7,80), (24,141), (138,1623)
 6084   4  (-12,66), (0,78), (13,91), (156,1950)
 6092   4  (-11,69), (-2,78), (14,94), (734,19886)
 6121   4  (-18,17), (-16,45), (230,3489), (995,31386)
 6137   4  (-8,75), (19,114), (38,247), (304,5301)
 6184   4  (-15,53), (-10,72), (6,80), (650,16572)
-6236   6  (20,42), (21,55), (60,458), (128,1446), (596,14550), (9368,906714)
 6372   5  (-11,71), (12,90), (48,342), (84,774), (829,23869)
 6400   7  (-16,48), (-15,55), (0,80), (20,120), (80,720), (96,944),
	   (10640,1097520)
-6400   4  (20,40), (40,240), (136,1584), (185,2515)
-6479   6  (20,39), (47,312), (80,711), (102,1027), (834,24085),
	   (6059,471630)
 6489   5  (-12,69), (18,111), (30,183), (58,449), (120,1317)
 6513   5  (-17,40), (16,103), (22,131), (1483,57110), (12652,1423111)
-6551   4  (26,105), (30,143), (368,7059), (3788,233139)
 6561   5  (-18,27), (0,81), (27,162), (54,405), (360,6831)
 6616   5  (-18,28), (-6,80), (105,1079), (362,6888), (494,10980)
 6625   6  (-10,75), (-4,81), (15,100), (24,143), (44,303), (690,18125)
-6691   4  (23,74), (155,1928), (695,18322), (77963,21768716)
 6713   5  (-14,63), (7,84), (8,85), (56,427), (28546,4823007)
-6823   4  (19,6), (28,123), (122,1345), (434,9041)
 6840   4  (-14,64), (6,84), (9,87), (3234,183912)
 6856   6  (-18,32), (-15,59), (30,184), (114,1220), (942,28912),
	   (27564105,144715764559)
-6908   5  (48,322), (93,893), (144,1726), (564,13394), (588,14258)
-6911   4  (20,33), (38,219), (138,1619), (9215,884592)
 6912   5  (-12,72), (-8,80), (24,144), (33,207), (1464,56016)
 6921   6  (-18,33), (10,89), (12,93), (34,215), (150,1839), (8259,750570)
 6940   4  (-19,9), (-6,82), (29,177), (74,642)
 7029   4  (-17,46), (3,84), (15,102), (6675,545352)
 7057  11  (-18,35), (-12,73), (-1,84), (8,87), (14,99), (48,343), (68,567),
	   (107,1110), (354,6661), (939,28774), (1674,68491)
 7084   4  (-19,15), (-10,78), (78,694), (40958,8289114)
 7100   4  (-14,66), (5,85), (10,90), (1210,42090)
-7100   5  (20,30), (24,82), (125,1395), (360,6830), (740,20130)
 7232   8  (-16,56), (-7,83), (8,88), (32,200), (44,304), (104,1064),
	   (1688,69352), (10049,1007359)
 7353   7  (-18,39), (-12,75), (6,87), (16,107), (66,543), (87,816),
	   (391,7732)
-7375   4  (20,25), (26,101), (395,7850), (536,12409)
 7388   5  (-19,23), (2,86), (26,158), (58,450), (277,4611)
-7424   5  (20,24), (24,80), (152,1872), (585,14149), (632,15888)
-7516   5  (20,22), (56,410), (68,554), (160,2022), (556,13110)
 7528   4  (6,88), (57,439), (3321,191383), (1664442,2147350804)
 7568   8  (-8,84), (-7,85), (1,87), (16,108), (56,428), (152,1876),
	   (368,7060), (28576,4830612)
-7600   4  (20,20), (161,2041), (500,11180), (860,25220)
-7740   4  (21,39), (24,78), (124,1378), (1476,56706)
-7775   5  (20,15), (26,99), (150,1835), (315,5590), (1560,61615)
 7785   8  (-14,71), (-9,84), (-6,87), (16,109), (24,147), (34,217),
	   (984,30867), (3264,186477)
-7804   7  (20,14), (32,158), (37,207), (92,878), (112,1182), (5260,381486),
	   (10405,1061361)
 7857   4  (-18,45), (4,89), (18,117), (11734,1271069)
-7900   4  (20,10), (44,278), (80,710), (3560,212410)
 7948   6  (-19,33), (-18,46), (-3,89), (102,1034), (206,2958), (6278,497430)
-7975   4  (20,5), (34,177), (115,1230), (14360,1720805)
 8004   4  (-20,2), (13,101), (16,110), (90040,27018002)
 8025   4  (-20,5), (10,95), (19,122), (14440,1735205)
 8036   4  (-20,6), (4,90), (32,202), (10040,1006006)
 8073   4  (-14,73), (3,90), (12,99), (426,8793)
 8100   5  (-20,10), (0,90), (36,234), (45,315), (3640,219610)
 8136   4  (-18,48), (-15,69), (82,748), (1554,61260)
 8225  11  (-20,15), (-10,85), (-5,90), (16,111), (50,365), (74,643),
	   (79,708), (130,1485), (1240,43665), (1640,66415),
	   (794740,708496335)
 8281   9  (-13,78), (0,91), (14,105), (39,260), (42,287), (140,1659),
	   (182,2457), (9282,894257), (41847,8560448)
 8289   7  (-20,17), (-12,81), (-2,91), (15,108), (46,325), (58,451), (96,945)
 8396   4  (-10,86), (22,138), (37,243), (85,789)
 8400   4  (-20,20), (4,92), (25,155), (940,28820)
 8433   6  (-18,51), (-8,89), (6,93), (48,345), (171,2238), (222,3309)
 8452   4  (-16,66), (-12,82), (44,306), (1397,52215)
-8532   7  (21,27), (22,46), (46,298), (102,1026), (318,5670), (541,12583),
	   (5286,384318)
 8569   4  (-10,87), (23,144), (36,235), (110781386,1166004406095)
-8623   5  (22,45), (68,553), (79,696), (178,2373), (704,18679)
 8673   6  (-14,77), (16,113), (22,139), (28,175), (532,12271), (1726,71707)
 8676   9  (-20,26), (-3,93), (12,102), (24,150), (60,474), (72,618),
	   (160,2026), (405,8151), (8149,735625)
-8712   6  (22,44), (33,165), (66,528), (81,723), (306,5352), (342,6324)
 8784   4  (-20,28), (-12,84), (81,735), (228,3444)
 8809   5  (3,94), (6,95), (18,121), (300,5197), (3960,249197)
 8828   4  (-14,78), (2,94), (13,105), (301,5223)
 8857   4  (-18,55), (-16,69), (83,762), (3878,241497)
 8900  12  (-20,30), (-11,87), (-4,94), (5,95), (16,114), (20,130), (40,270),
	   (200,2830), (340,6270), (440,9230), (3685,223695), (23245,3544005)
 8961   4  (-20,31), (-5,94), (4,95), (442,9293)
 9000   5  (-15,75), (6,96), (10,100), (54,408), (2385,116475)
 9024   5  (-20,32), (1,95), (25,157), (28,176), (12268,1358816)
 9052   5  (-6,94), (-3,95), (18,122), (69,581), (3978,250898)
 9100   5  (-10,90), (29,183), (30,190), (2094,95822), (139070,51862110)
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 9280   5  (-16,72), (-4,96), (24,152), (216,3176), (220289,103392543)
 9284   4  (5,97), (56,430), (80,722), (2168,100946)
 9297   8  (-21,6), (-12,87), (12,105), (18,123), (43,298), (64,521),
	   (114,1221), (1414,53171)
-9408   4  (28,112), (49,329), (364,6944), (18529,2522191)
 9432   4  (-18,60), (-6,96), (33,213), (193,2683)
-9455   4  (126,1411), (171,2234), (924,28087), (6264,495767)
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	   (3622,217983)
 9585   4  (-21,18), (6,99), (24,153), (166,2141)
 9649   4  (-12,89), (-10,93), (26,165), (8303,756576)
 9676   4  (-18,62), (5,99), (62,498), (125,1401)
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	   (614,15214), (8069,724819), (12997,1481715)
 9800   4  (-14,84), (1,99), (14,112), (49,357)
 9801   4  (-18,63), (0,99), (22,143), (99,990)
 9828   4  (-12,90), (-3,99), (16,118), (456,9738)
 9836   4  (-10,94), (29,185), (70,594), (262,4242)
 9936   6  (-20,44), (-15,81), (4,100), (12,108), (52,388), (660,16956)
-9967   6  (32,151), (56,407), (98,965), (1052,34121), (72971,19711762),
	   (195467,86419186)

© Helmut Richter      published here 2001-03-15; last update 2001-04-02      http://www.lrz.de/~hr/numb/mordell.html