Unique solutions (using squares only) to Pillai's equation for n <= 1000 Perfect powers examined up to 2^63-1 n for which there is a unique solution in squares: 29, 43, 52, 59, 173, 181, 263, 283, 317, 332, 347, 349, 361, 379, 383, 419, 428, 436, 461, 467, 479, 484, 491, 509, 523, 529, 569, 571, 607, 613, 619, 641, 643, 653, 661, 677, 691, 709, 733, 773, 787, 788, 811, 827, 839, 853, 877, 881, 883, 907, 911, 941, 947, 956, 983 Total number of solutions: 55 Solutions for each n: 29 = 15^2 - 14^2 43 = 22^2 - 21^2 52 = 14^2 - 12^2 59 = 30^2 - 29^2 173 = 87^2 - 86^2 181 = 91^2 - 90^2 263 = 132^2 - 131^2 283 = 142^2 - 141^2 317 = 159^2 - 158^2 332 = 84^2 - 82^2 347 = 174^2 - 173^2 349 = 175^2 - 174^2 361 = 181^2 - 180^2 379 = 190^2 - 189^2 383 = 192^2 - 191^2 419 = 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 - 120^2 491 = 246^2 - 245^2 509 = 255^2 - 254^2 523 = 262^2 - 261^2 529 = 265^2 - 264^2 569 = 285^2 - 284^2 571 = 286^2 - 285^2 607 = 304^2 - 303^2 613 = 307^2 - 306^2 619 = 310^2 - 309^2 641 = 321^2 - 320^2 643 = 322^2 - 321^2 653 = 327^2 - 326^2 661 = 331^2 - 330^2 677 = 339^2 - 338^2 691 = 346^2 - 345^2 709 = 355^2 - 354^2 733 = 367^2 - 366^2 773 = 387^2 - 386^2 787 = 394^2 - 393^2 788 = 198^2 - 196^2 811 = 406^2 - 405^2 827 = 414^2 - 413^2 839 = 420^2 - 419^2 853 = 427^2 - 426^2 877 = 439^2 - 438^2 881 = 441^2 - 440^2 883 = 442^2 - 441^2 907 = 454^2 - 453^2 911 = 456^2 - 455^2 941 = 471^2 - 470^2 947 = 474^2 - 473^2 956 = 240^2 - 238^2 983 = 492^2 - 491^2