Table 1. Number of known solutions of y² = x³ + k for 0 < k < 1008
Table 2. For 0 < k < 1008: smallest x such that x³ + k is a square
Table 3. Number of known solutions of y² = x³ - k for 0 < k < 1008
Table 4. For 0 < k < 1008: smallest x such that x³ - k is a square
Table 6. Solutions of y² = x³ + k for 1008 <= | k | <= 10000
Big Table: one file with everything (plain text, compressed)
This article contains in several tables all solutions to Mordell's equation y² = x³ + 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".
Here, for given k, one solution is one value of x so that x³ + 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
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 . . .
Here, for given k, one solution is one value of x so that x³ - 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
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
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)
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)
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