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