Be sure to e-mail me if you find any new binary/decimal palindromes!
term - bitlen - binary representation - decimal representation 1 1 0 0 2 1 1 1 3 2 11 3 4 3 101 5 5 3 111 7 6 4 1001 9 7 6 100001 33 8 7 1100011 99 9 9 100111001 313 10 10 1001001001 585 11 10 1011001101 717 12 13 1110100010111 7447 13 14 10001100110001 9009 14 14 11101111110111 15351 15 15 111110111011111 32223 16 16 1001110000111001 39993 17 16 1100111111110011 53235 18 16 1101001001001011 53835 19 17 10010000000001001 73737 20 20 10001110111101110001 585585 21 21 110101101010101101011 1758571 22 21 111011000010000110111 1934391 23 21 111100011010110001111 1979791 24 22 1011111011111101111101 3129213 25 23 10011010110001101011001 5071705 26 23 10100000100000100000101 5259525 27 23 10110010010001001001101 5841485 28 24 110011100000000001110011 13500531 29 30 101010111010000000010111010101 719848917 30 30 110110010000110011000010011011 910373019 31 30 110111111111110011111111111011 939474939 32 31 1001100111100010100011110011001 1290880921 33 33 110111100000111101111000001111011 7451111547 34 34 1001010111000101001010001110101001 10050905001 35 35 10001001100011011011011000110010001 18462126481 36 35 11110001111111010101011111110001111 32479297423 37 37 1000101110111010000000101110111010001 75015151057 38 37 1100111010101000100010001010101110011 110948849011 39 37 1111111001001100011100011001001111111 136525525631 40 41 10001111101010110010101001101010111110001 1234104014321 41 41 10100100100110011000000011001100100100101 1413899983141 42 41 10101011101101000001110000010110111010101 1474922294741 43 41 11010000101100101011111010100110100001011 1792704072971 44 41 11010000110111000100000100011101100001011 1794096904971 45 41 11101000110100100110101100100101100010111 1999925299991 46 43 1010010010000011010100010101100000100100101 5652622262565 47 43 1101001001011001010001000101001101001001011 7227526257227 48 43 1101010000000011010111110101100000000101011 7284717174827 49 44 10001010000001011110100101111010000001010001 9484874784849 50 45 111110000110100101001010100101001011000011111 34141388314143 51 49 1111101100011110000000000000000000111100011011111 552212535212255 52 50 11010100001010111100110100101100111101010000101011 933138363831339 53 51 110010111110110110011001101100110011011011111010011 1793770770773971 54 52 1011001011111111010101100110011010101111111101001101 3148955775598413 55 54 100101001001110001111011010010110111100011100100101001 10457587478575401 56 54 100110011100000110111100101101001111011000001110011001 10819671917691801 57 55 1000000111100010000110111110111110110000100011110000001 18279440804497281 58 55 1111001001010011101100010000000100011011100101001001111 34104482028440143 59 56 10000011101110101111100000100100000111110101110111000001 37078796869787073 60 56 10000101101100000011100000100100000111000000110110100001 37629927072992673 61 56 11000110110010001010000010000001000001010001001101100011 55952637073625955 62 58 1000111100101110000010010000110000100100000111010011110001 161206152251602161 63 59 10001011001111110110111101010101010111101101111110011010001 313558153351855313 64 63 110000110100101110101111010101111101010111101011101001011000011 7036267126217626307 65 64 1000011001000100000010011101100000011011100100000010001001100001 9674868723278684769 66 64 1001011011111100100110100001101111011000010110010011111101101001 10879740244204797801 67 65 11100100001110000100010111100110001100111101000100001110000100111 32889941788714998823 68 67 1010010001101001111101010000000010100000000101011111001011000100101 94778157422475187749 69 67 1100011001011111011011101111101010101011111011101101111101001100011 114354126121621453411 70 69 111010011111010111001011000001011010110100000110100111010111110010111 539475328171823574935 71 74 11101100101001110111101010100001100011000110000101010111101110010100110111 17461998948684989916471 72 76 1111011100010111000010001000100000001001000000010001000100001110100011101111 72928088195859188082927 73 77 10011101011001101100001111001011011101110111011010011110000110110011010111001 92913401775957710431929 74 77 10011111101011111000101010110000101100000110100001101010100011111010111111001 94261805583838550816249 75 80 11101111011000111100010111011101101100100100110110111011101000111100011011110111 1130486074817184706840311 76 80 11111100111001111110000011110100111000100100011100101111000001111110011100111111 1194313761393931673134911 77 81 100000010001011100111110110101100111111101111111001101011011111001110100010000001 1219228158701078518229121 78 83 10011001110111100101011110111001110100001110000101110011101111010100111101110011001 5812988563013103658892185 79 83 11000000001100100101111110001001011010100100101011010010001111110100100110000000011 7260988688520258868890627 80 83 11000101111000010101010110100001110100000100000101110000101101010101000011110100011 7475703079870789703075747 81 86 10101000001010100000100110101010101001100000000110010101010101100100000101010000010101 50824513851188115831542805 82 87 111001011000111001101111010110010111011100111001110111010011010111101100111000110100111 138758321383797383123857831 83 89 10010001001101011010101000000110111100000100000100000111101100000010101011010110010001001 351095331428353824133590153 84 89 10100001100110001000000111100001100011000111011100011000110000111100000010001100110000101 390714505091666190505417093 85 89 11011100000100000001001010010000011001110101010101110011000001001010010000000100000111011 532079161251434152161970235 86 89 11100100101100110101011000010001001111100100100100111110010001000011010101100110100100111 552963956270141072659369255 87 90 101001000111010111001110010101001010000101000000101000010100101010011100111010111000100101 795280629691202196926082597 88 91 1010011001011111011111100001110110010000000100010000000100110111000011111101111101001100101 1609061098335005338901609061 89 94 1110011011110110001110101001000001000000000011001100000000001000001001010111000110111101100111 17869806142184248124160896871 90 94 1111111101011000000101011001100101101101100001111000011011011010011001101010000001101011111111 19756291244127372144219265791 91 95 11000001110111110100001110001010010000110100011011000101100001001010001110000101111101110000011 30000258151173237115185200003 92 95 11000110001000000010110011101000100010000101111111110100001000100010111001101000000010001100011 30658464822225352222846485603 93 96 101101101010101001110101110101101111011010100000000001010110111101101011101011100101010101101101 56532345659072227095654323565 94 99 100110001011001001110111010000000001110111000111010111000111011100000000010111011100100110100011001 378059787464677776464787950873 95 100 1110000101010101000110001001111111101011111110110110110111111101011111111001000110001010101010000111 1115792035060833380605302975111 96 102 101010110011000001001111000000100001000111101001011110100101111000100001000000111100100000110011010101 3390741646331381831336461470933 97 103 1011010010100010100000100100110111011100110000100000000010000110011101110110010010000010100010100101101 7155681676104835384016761865517 98 108 100101001110000100000011000000011101011010000111111001100111111000010110101110000000110000001000011100101001 188726493036450333054630394627881 99 109 1000001101011000000111101011111010011111110010111111110111111110100111111100101111101011110000001101011000001 332997156422555464555224651799233 100 109 1110010010101111000010000000101101110101110101111110010100111111010111010111011010000000100001111010100100111 579782100975917393719579001287975 101 111 100010101111101111001011011110110000100110111101111101111101111101111011001000011011110110100111101111101010001 1409460884147943003497414880649041 102 111 100100100000011001100111010000100000111000101000010010101010010000101000111000001000010111001100110000001001001 1480869563960100770010693659680841 103 113 11110100110110110000110000111111001110101110110100100110101100100101101110101110011111100001100001101101100101111 9932525402284695775964822045252399 104 114 100000110100110000011001000011010100110111001111111010100001010111111100111011001010110000100110000011001011000001 10652099006552766666725560099025601 105 114 100001010111010111111000100101110101000001101110111000000000000111011101100000101011101001000111111010111010100001 10827628430039640604693003482672801 106 116 10100110101000011001101101000110100100010010100111010001000010001011100101001000100101100010110110011000010101100101 54074940541725088788052714504947045 107 116 11101100100010001100100111000111101110000000111101011110111101111010111100000001110111100011100100110001000100110111 76759778311938325452383911387795767 108 117 100100010100010100110101110000011000101111001111001010100101001010100111100111101000110000011101011001010001010001001 94285848717805140304150871784858249 109 117 100101010001001100000010001001011010110010111110000001011010110100000011111010011010110100100010000001100100010101001 96754720977532710701723577902745769 110 117 110101100101101011100001111100010101001010100000101110011010110011101000001010100101010001111100001110101101001101011 139124355701640720027046107553421931 111 120 100010001100000111011011000001010110000011110100110011101001100101110011001011110000011010100000110110111000001100010001 710084230446469950059964644032480017 112 120 100110001111111011010110010010110011100000110100011011010110011010110110001011000001110011010010011010110111111100011001 794397832642722540045227246238793497 113 120 110000000011011000111101011111110111001001111010100001010011110010100001010111100100111011111110101111000110110000000011 998021119318189842248981813911120899 114 120 111111101110010001100000110000001011101110000100010011011001100110110010001000011101110100000011000001100010011101111111 1323475457008895965695988007545743231 115 121 1010000111110100001011100011010010000100000001001100100000111110000010011001000000010000100101100011101000010111110000101 1681824725831390428240931385274281861 116 123 100011011110001111100110111011011000000011010001101100111101101101111001101100010110000000110110111011001111100011110110001 5893890080115984244424895110800983985 117 123 111010011011011000101001111110111011010000110111010111101000111000101111010111011000010110111011111100101000110110110010111 9707999142717984907094897172419997079 118 123 111100000000011101000001101111110000110101100111000001101101101101101100000111001101011000011111101100000101110000000001111 9970387454991896491946981994547830799 119 125 11000001101111001100101010101110011101011101101110101100001011011010000110101110110111010111001110101010100110011110110000011 32190158233101105022050110133285109123 120 127 1011011111101101011100111110001001111001110001100111000010000101010000100001110011000111001111001000111110011101011011111101101 122240824002234545959545432200428042221 121 127 1011101011011101001011100011110110110011001011010010000001000110110001000000100101101001100110110111100011101001011101101011101 124192421350471300727003174053124291421 122 127 1100011000011111001001111110101101011010100110011111011010111000001110101101111100110010101101011010111111001001111100001100011 131674457014330218696812033410754476131 123 129 110011111000011011001000100100101000110001100100000110010010100101001010010011000001001100011000101001001000100110110000111110011 551700061998405245575542504899160007155 124 130 1001000101111111111100101011111011001100011000110110000111111000000001111110000110110001100011001101111101010011111111111010001001 773609618198307097595790703891816906377 125 130 1011111101011011000100110111110111011110111111100010000000011111111111100000000100011111110111101110111110110010001101101011111101 1017421766189445102992015449816671247101 126 131 10011001101101110100000100001000001100011100101110011110001110100000101110001111001110100111000110000010000100000101110110110011001 1634587141488515712882175158841417854361 127 133 1010011101000101010010110110101101011101000100000110001101110010110110100111011000110000010001011101011010110110100101010001011100101 7114907950920173924554293710290597094117 128 134 10101000011010110000011011100110011010100011111101011110100110011100111001100101111010111111000101011001100111011000001101011000010101 14327425216354951264146215945361252472341 129 135 101110011111000110110100100110000011011010101010010000010001011110010011110100010000010010101010110110000011001001011011000111110011101 31636759764024794204540249742046795763613
This is the first algorythm, but a distributed-computing version of it where ranges of binary-counter spaces are checked on various distributed computers.
Here is the current algorythm that I am using. It uses a more sophisticated way of eliminating search-spaces based on a few mathematical rules. Basically, it generates the decimal palindrome first, incrementing it instead of the binary counter. As it translates the decimal palindrome to binary, it aborts the process if it sees it's impossible for the binary translation to be palindromic. It might seem counterintuitive to achieve more performance by starting with the decimal and converting to binary rather than the other way around, but it actually increases the search speed quite a bit!
Related Links:
World of Numbers website
Contact me at: charlton@dynet.com
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