Solutions to Pillai's equation for n <= 100 Perfect powers examined up to 10^18 Solution frequency: 1,1,2,3,2,0,5,3,4,1,4,2,3,0,3,3,7,3,5,2,2,2,4,5,2,3,3,7,1,1,2,4,2,0,3,2,3,1,4,4, 3,0,1,3,4,1,6,4,3,0,2,1,2,2,3,4,3,0,1,4,2,0,4,4,4,0,2,5,2,0,4,4,6,2,3,3,2,0,4,4, 4,0,2,2,2,0,3,3,6,0,3,4,4,2,4,5,3,2,4,10 Total number of solutions: 274 Solutions for each n: 1 = 3^2 - 2^3 2 = 3^3 - 5^2 3 = 2^2 - 1^2 = 2^7 - 5^3 4 = 2^3 - 2^2 = 6^2 - 2^5 = 5^3 - 11^2 5 = 3^2 - 2^2 = 2^5 - 3^3 6 none 7 = 2^3 - 1^2 = 4^2 - 3^2 = 2^5 - 5^2 = 2^7 - 11^2 = 32^3 - 181^2 8 = 3^2 - 1^2 = 4^2 - 2^3 = 312^2 - 46^3 9 = 5^2 - 4^2 = 6^2 - 3^3 = 15^2 - 6^3 = 253^2 - 40^3 10 = 13^3 - 3^7 11 = 3^3 - 4^2 = 6^2 - 5^2 = 56^2 - 5^5 = 15^3 - 58^2 12 = 4^2 - 2^2 = 47^2 - 13^3 13 = 7^2 - 6^2 = 16^2 - 3^5 = 17^3 - 70^2 14 none 15 = 4^2 - 1^2 = 8^2 - 7^2 = 1138^2 - 109^3 16 = 5^2 - 3^2 = 2^5 - 4^2 = 12^2 - 2^7 17 = 5^2 - 2^3 = 7^2 - 2^5 = 9^2 - 8^2 = 23^2 - 8^3 = 282^2 - 43^3 = 375^2 - 52^3 = 378661^2 - 5234^3 18 = 3^3 - 3^2 = 3^5 - 15^2 = 19^2 - 7^3 19 = 3^3 - 2^3 = 10^2 - 9^2 = 12^2 - 5^3 = 7^3 - 18^2 = 55^5 - 22434^2 20 = 6^2 - 4^2 = 6^3 - 14^2 21 = 5^2 - 2^2 = 11^2 - 10^2 22 = 7^2 - 3^3 = 47^2 - 3^7 23 = 3^3 - 2^2 = 2^5 - 3^2 = 12^2 - 11^2 = 2^11 - 45^2 24 = 5^2 - 1^2 = 2^5 - 2^3 = 7^2 - 5^2 = 32^2 - 10^3 = 736844^2 - 8158^3 25 = 5^3 - 10^2 = 13^2 - 12^2 26 = 3^3 - 1^2 = 35^3 - 207^2 = 2537^2 - 23^5 27 = 6^2 - 3^2 = 14^2 - 13^2 = 3^5 - 6^3 28 = 2^5 - 2^2 = 6^2 - 2^3 = 8^2 - 6^2 = 2^7 - 10^2 = 8^3 - 22^2 = 37^3 - 225^2 = 2^17 - 362^2 29 = 15^2 - 14^2 30 = 83^2 - 19^3 31 = 2^5 - 1^2 = 16^2 - 15^2 32 = 6^2 - 2^2 = 8^2 - 2^5 = 9^2 - 7^2 = 6^5 - 88^2 33 = 7^2 - 4^2 = 17^2 - 16^2 34 none 35 = 6^2 - 1^2 = 18^2 - 17^2 = 11^3 - 36^2 36 = 10^2 - 8^2 = 42^2 - 12^3 37 = 8^2 - 3^3 = 19^2 - 18^2 = 3788^2 - 243^3 38 = 37^2 - 11^3 39 = 8^2 - 5^2 = 20^2 - 19^2 = 10^3 - 31^2 = 22^3 - 103^2 40 = 7^2 - 3^2 = 11^2 - 9^2 = 16^2 - 6^3 = 14^3 - 52^2 41 = 7^2 - 2^3 = 13^2 - 2^7 = 21^2 - 20^2 42 none 43 = 22^2 - 21^2 44 = 5^3 - 9^2 = 12^2 - 10^2 = 13^2 - 5^3 45 = 7^2 - 2^2 = 9^2 - 6^2 = 23^2 - 22^2 = 21^3 - 96^2 46 = 17^2 - 3^5 47 = 2^7 - 9^2 = 6^3 - 13^2 = 3^5 - 14^2 = 24^2 - 23^2 = 12^3 - 41^2 = 63^3 - 500^2 48 = 7^2 - 1^2 = 8^2 - 4^2 = 13^2 - 11^2 = 28^3 - 148^2 49 = 9^2 - 2^5 = 25^2 - 24^2 = 65^3 - 524^2 50 none 51 = 10^2 - 7^2 = 26^2 - 25^2 52 = 14^2 - 12^2 53 = 27^2 - 26^2 = 29^3 - 156^2 54 = 9^2 - 3^3 = 7^3 - 17^2 55 = 8^2 - 3^2 = 28^2 - 27^2 = 56^3 - 419^2 56 = 8^2 - 2^3 = 9^2 - 5^2 = 15^2 - 13^2 = 18^3 - 76^2 57 = 11^2 - 8^2 = 20^2 - 7^3 = 29^2 - 28^2 58 none 59 = 30^2 - 29^2 60 = 8^2 - 2^2 = 16^2 - 14^2 = 136^3 - 1586^2 = 76^5 - 50354^2 61 = 5^3 - 8^2 = 31^2 - 30^2 62 none 63 = 8^2 - 1^2 = 12^2 - 9^2 = 32^2 - 31^2 = 568^3 - 13537^2 64 = 10^2 - 6^2 = 2^7 - 8^2 = 17^2 - 15^2 = 24^2 - 8^3 65 = 9^2 - 4^2 = 33^2 - 32^2 = 53^2 - 14^3 = 14113^2 - 584^3 66 none 67 = 34^2 - 33^2 = 23^3 - 110^2 68 = 10^2 - 2^5 = 14^2 - 2^7 = 18^2 - 16^2 = 46^2 - 2^11 = 1874^2 - 152^3 69 = 13^2 - 10^2 = 35^2 - 34^2 70 none 71 = 14^2 - 5^3 = 8^3 - 21^2 = 36^2 - 35^2 = 3^7 - 46^2 72 = 9^2 - 3^2 = 11^2 - 7^2 = 6^3 - 12^2 = 19^2 - 17^2 73 = 9^2 - 2^3 = 10^2 - 3^3 = 17^2 - 6^3 = 37^2 - 36^2 = 611^2 - 72^3 = 6717^2 - 356^3 74 = 3^5 - 13^2 = 99^3 - 985^2 75 = 10^2 - 5^2 = 14^2 - 11^2 = 38^2 - 37^2 76 = 5^3 - 7^2 = 20^2 - 18^2 = 101^3 - 1015^2 77 = 9^2 - 2^2 = 39^2 - 38^2 78 none 79 = 2^7 - 7^2 = 40^2 - 39^2 = 20^3 - 89^2 = 302^2 - 45^3 80 = 9^2 - 1^2 = 12^2 - 8^2 = 21^2 - 19^2 = 292^2 - 44^3 81 = 15^2 - 12^2 = 18^2 - 3^5 = 41^2 - 40^2 = 13^3 - 46^2 82 none 83 = 42^2 - 41^2 = 27^3 - 140^2 84 = 10^2 - 4^2 = 22^2 - 20^2 85 = 11^2 - 6^2 = 43^2 - 42^2 86 none 87 = 16^2 - 13^2 = 7^3 - 16^2 = 44^2 - 43^2 88 = 13^2 - 9^2 = 6^3 - 2^7 = 23^2 - 21^2 89 = 11^2 - 2^5 = 5^3 - 6^2 = 33^2 - 10^3 = 45^2 - 44^2 = 91^2 - 2^13 = 408^2 - 55^3 90 none 91 = 10^2 - 3^2 = 6^3 - 5^3 = 46^2 - 45^2 92 = 10^2 - 2^3 = 2^7 - 6^2 = 24^2 - 22^2 = 2^13 - 90^2 93 = 5^3 - 2^5 = 17^2 - 14^2 = 47^2 - 46^2 = 130^2 - 7^5 94 = 11^2 - 3^3 = 421^2 - 3^11 95 = 12^2 - 7^2 = 6^3 - 11^2 = 48^2 - 47^2 = 6^7 - 529^2 96 = 10^2 - 2^2 = 11^2 - 5^2 = 2^7 - 2^5 = 14^2 - 10^2 = 25^2 - 23^2 97 = 15^2 - 2^7 = 49^2 - 48^2 = 77^2 - 18^3 98 = 5^3 - 3^3 = 21^2 - 7^3 99 = 10^2 - 1^2 = 3^5 - 12^2 = 18^2 - 15^2 = 50^2 - 49^2 100 = 5^3 - 5^2 = 15^2 - 5^3 = 7^3 - 3^5 = 26^2 - 24^2 = 10^3 - 30^2 = 5^5 - 55^2 = 90^2 - 20^3 = 118^2 - 24^3 = 34^3 - 198^2 = 137190^2 - 2660^3