Unique solutions to Pillai's equation for n <= 1000 Perfect powers examined up to 2^63-1 n for which there is a unique solution: 1, 2, 10, 29, 30, 38, 43, 46, 52, 59, 122, 126, 138, 142, 146, 150, 154, 166, 170, 173, 181, 190, 194, 214, 222, 234, 263, 270, 282, 283, 298, 317, 318, 332, 338, 342, 347, 349, 354, 361, 370, 379, 382, 383, 386, 406, 419, 428, 436, 461, 467, 479, 484, 486, 490, 491, 498, 502, 509, 518, 523, 529, 546, 550, 566, 569, 571, 574, 582, 586, 594, 607, 613, 619, 638, 641, 643, 653, 661, 666, 677, 678, 686, 691, 694, 709, 710, 726, 730, 733, 734, 746, 773, 774, 787, 788, 806, 811, 814, 818, 827, 834, 838, 839, 853, 877, 878, 881, 883, 894, 906, 907, 911, 930, 934, 938, 941, 947, 956, 962, 970, 974, 982, 983 Total number of solutions: 124 Solutions for each n: 1 = 3^2 - 2^3 2 = 3^3 - 5^2 10 = 13^3 - 3^7 29 = 15^2 - 14^2 30 = 83^2 - 19^3 38 = 37^2 - 11^3 43 = 22^2 - 21^2 46 = 17^2 - 3^5 52 = 14^2 - 12^2 59 = 30^2 - 29^2 122 = 3^5 - 11^2 126 = 15^3 - 57^2 138 = 173^2 - 31^3 142 = 13^2 - 3^3 146 = 195^3 - 2723^2 150 = 175^3 - 2315^2 154 = 111^2 - 23^3 166 = 7^5 - 129^2 170 = 59^3 - 453^2 173 = 87^2 - 86^2 181 = 91^2 - 90^2 190 = 39^2 - 11^3 194 = 3^5 - 7^2 214 = 49^2 - 3^7 222 = 7^3 - 11^2 234 = 3^5 - 3^2 263 = 132^2 - 131^2 270 = 39^3 - 9^5 282 = 25^2 - 7^3 283 = 142^2 - 141^2 298 = 19^3 - 81^2 317 = 159^2 - 158^2 318 = 7^3 - 5^2 332 = 84^2 - 82^2 338 = 3^7 - 43^2 342 = 7^3 - 1^2 347 = 174^2 - 173^2 349 = 175^2 - 174^2 354 = 131^2 - 7^5 361 = 181^2 - 180^2 370 = 11^3 - 31^2 379 = 190^2 - 189^2 382 = 25^2 - 3^5 383 = 192^2 - 191^2 386 = 9^3 - 7^3 406 = 1107^2 - 107^3 419 = 210^2 - 209^2 428 = 108^2 - 106^2 436 = 110^2 - 108^2 461 = 231^2 - 230^2 467 = 234^2 - 233^2 479 = 240^2 - 239^2 484 = 122^2 - 120^2 486 = 9^3 - 3^5 490 = 11^3 - 29^2 491 = 246^2 - 245^2 498 = 29^2 - 7^3 502 = 23^2 - 3^3 509 = 255^2 - 254^2 518 = 43^2 - 11^3 523 = 262^2 - 261^2 529 = 265^2 - 264^2 546 = 43^3 - 281^2 550 = 31^3 - 171^2 566 = 15^3 - 53^2 569 = 285^2 - 284^2 571 = 286^2 - 285^2 574 = 365^2 - 51^3 582 = 283^2 - 43^3 586 = 115^3 - 1233^2 594 = 63^2 - 15^3 607 = 304^2 - 303^2 613 = 307^2 - 306^2 619 = 310^2 - 309^2 638 = 549^2 - 67^3 641 = 321^2 - 320^2 643 = 322^2 - 321^2 653 = 327^2 - 326^2 661 = 331^2 - 330^2 666 = 3^7 - 39^2 677 = 339^2 - 338^2 678 = 7^5 - 127^2 686 = 2395^2 - 179^3 691 = 346^2 - 345^2 694 = 45^2 - 11^3 709 = 355^2 - 354^2 710 = 87^2 - 19^3 726 = 55^3 - 407^2 730 = 3511^2 - 231^3 733 = 367^2 - 366^2 734 = 15^5 - 871^2 746 = 33^2 - 7^3 773 = 387^2 - 386^2 774 = 15^3 - 51^2 787 = 394^2 - 393^2 788 = 198^2 - 196^2 806 = 209^2 - 35^3 811 = 406^2 - 405^2 814 = 29^2 - 3^3 818 = 3^7 - 37^2 827 = 414^2 - 413^2 834 = 175^2 - 31^3 838 = 55^2 - 3^7 839 = 420^2 - 419^2 853 = 427^2 - 426^2 877 = 439^2 - 438^2 878 = 47^2 - 11^3 881 = 441^2 - 440^2 883 = 442^2 - 441^2 894 = 7^7 - 907^2 906 = 409^2 - 55^3 907 = 454^2 - 453^2 911 = 456^2 - 455^2 930 = 19^3 - 77^2 934 = 31^2 - 3^3 938 = 5^5 - 3^7 941 = 471^2 - 470^2 947 = 474^2 - 473^2 956 = 240^2 - 238^2 962 = 3^7 - 35^2 970 = 11^3 - 19^2 974 = 15^3 - 49^2 982 = 35^2 - 3^5 983 = 492^2 - 491^2