Powers of 2 with no Zero in Decimal Representation

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Unproven Hypotheses

The following powers of $2$ which contain no zero in their decimal representation are believed to be all that exist:

$1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 8192, 16 \, 384,$
$32 \, 768, 65 \, 536, 262 \, 144, 524 \, 288, 16 \, 777 \, 216, 33 \, 554 \, 432, 134 \, 217 \, 728,$
$268 \, 435 \, 456, 2 \, 147 \, 483 \, 648, 4 \, 294 \, 967 \, 296, 8 \, 589 \, 934 \, 592, 17 \, 179 \, 869 \, 184,$
$34 \, 359 \, 738 \, 368, 68 \, 719 \, 476 \, 736, 137 \, 438 \, 953 \, 472, 549 \, 755 \, 813 \, 888,$
$562 \, 949 \, 953 \, 421 \, 312, 2 \, 251 \, 799 \, 813 \, 685 \, 248, 147 \, 573 \, 952 \, 589 \, 676 \, 412 \, 928,$
$4 \, 722 \, 366 \, 482 \, 869 \, 645 \, 213 \, 696, 75 \, 557 \, 863 \, 725 \, 914 \, 323 \, 419 \, 136, 151 \, 115 \, 727 \, 451 \, 828 \, 646 \, 838 \, 272,$
$2 \, 417 \, 851 \, 639 \, 229 \, 258 \, 349 \, 412 \, 352, 77 \, 371 \, 252 \, 455 \, 336 \, 267 \, 181 \, 195 \, 264$

but this has not been proven.

This sequence is A238938 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


The corresponding indices are:

$0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 13, 14, 15, 16, 18, 19, 24, 25, 27, 28, 31, 32, 33, 34, 35, 36, 37, 39, 49, 51, 67, 72, 76, 77, 81, 86$

This sequence is A007377 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Progress

\(\displaystyle 2^0\) \(=\) \(\displaystyle 1\)
\(\displaystyle 2^1\) \(=\) \(\displaystyle 2\)
\(\displaystyle 2^2\) \(=\) \(\displaystyle 4\)
\(\displaystyle 2^3\) \(=\) \(\displaystyle 8\)
\(\displaystyle 2^4\) \(=\) \(\displaystyle 16\)
\(\displaystyle 2^5\) \(=\) \(\displaystyle 32\)
\(\displaystyle 2^6\) \(=\) \(\displaystyle 64\)
\(\displaystyle 2^7\) \(=\) \(\displaystyle 128\)
\(\displaystyle 2^9\) \(=\) \(\displaystyle 512\)
\(\displaystyle 2^{10}\) \(=\) \(\displaystyle 1024\) which contains a zero
\(\displaystyle 2^{11}\) \(=\) \(\displaystyle 2048\) which contains a zero
\(\displaystyle 2^{12}\) \(=\) \(\displaystyle 4096\) which contains a zero
\(\displaystyle 2^{13}\) \(=\) \(\displaystyle 8192\)
\(\displaystyle 2^{14}\) \(=\) \(\displaystyle 16 \, 384\)
\(\displaystyle 2^{15}\) \(=\) \(\displaystyle 32 \, 768\)
\(\displaystyle 2^{16}\) \(=\) \(\displaystyle 65 \, 536\)
\(\displaystyle 2^{17}\) \(=\) \(\displaystyle 131 \, 072\) which contains a zero
\(\displaystyle 2^{18}\) \(=\) \(\displaystyle 262 \, 144\)
\(\displaystyle 2^{19}\) \(=\) \(\displaystyle 524 \, 288\)
\(\displaystyle 2^{20}\) \(=\) \(\displaystyle 1 \, 048 \, 576\) which contains a zero
\(\displaystyle 2^{21}\) \(=\) \(\displaystyle 2 \, 097 \, 152\) which contains a zero
\(\displaystyle 2^{22}\) \(=\) \(\displaystyle 4 \, 194 \, 304\) which contains a zero
\(\displaystyle 2^{23}\) \(=\) \(\displaystyle 8 \, 388 \, 608\) which contains a zero
\(\displaystyle 2^{24}\) \(=\) \(\displaystyle 16 \, 777 \, 216\)
\(\displaystyle 2^{25}\) \(=\) \(\displaystyle 33 \, 554 \, 432\)
\(\displaystyle 2^{26}\) \(=\) \(\displaystyle 67 \, 108 \, 864\) which contains a zero
\(\displaystyle 2^{27}\) \(=\) \(\displaystyle 134 \, 217 \, 728\)
\(\displaystyle 2^{28}\) \(=\) \(\displaystyle 268 \, 435 \, 456\)
\(\displaystyle 2^{29}\) \(=\) \(\displaystyle 536 \, 870 \, 912\) which contains a zero
\(\displaystyle 2^{30}\) \(=\) \(\displaystyle 1 \, 073 \, 741 \, 824\) which contains a zero
\(\displaystyle 2^{31}\) \(=\) \(\displaystyle 2 \, 147 \, 483 \, 648\)
\(\displaystyle 2^{32}\) \(=\) \(\displaystyle 4 \, 294 \, 967 \, 296\)
\(\displaystyle 2^{33}\) \(=\) \(\displaystyle 8 \, 589 \, 934 \, 592\)
\(\displaystyle 2^{34}\) \(=\) \(\displaystyle 17 \, 179 \, 869 \, 184\)
\(\displaystyle 2^{35}\) \(=\) \(\displaystyle 34 \, 359 \, 738 \, 368\)
\(\displaystyle 2^{36}\) \(=\) \(\displaystyle 68 \, 719 \, 476 \, 736\)
\(\displaystyle 2^{37}\) \(=\) \(\displaystyle 137 \, 438 \, 953 \, 472\)
\(\displaystyle 2^{38}\) \(=\) \(\displaystyle 274 \, 877 \, 906 \, 944\) which contains a zero
\(\displaystyle 2^{39}\) \(=\) \(\displaystyle 549 \, 755 \, 813 \, 888\)
\(\displaystyle 2^{40}\) \(=\) \(\displaystyle 1 \, 099 \, 511 \, 627 \, 776\) which contains a zero
\(\displaystyle 2^{41}\) \(=\) \(\displaystyle 2 \, 199 \, 023 \, 255 \, 552\) which contains a zero
\(\displaystyle 2^{42}\) \(=\) \(\displaystyle 4 \, 398 \, 046 \, 511 \, 104\) which contains zeroes
\(\displaystyle 2^{43}\) \(=\) \(\displaystyle 8 \, 796 \, 093 \, 022 \, 208\) which contains zeroes
\(\displaystyle 2^{44}\) \(=\) \(\displaystyle 17 \, 592 \, 186 \, 044 \, 416\) which contains a zero
\(\displaystyle 2^{45}\) \(=\) \(\displaystyle 35 \, 184 \, 372 \, 088 \, 832\) which contains a zero
\(\displaystyle 2^{46}\) \(=\) \(\displaystyle 70 \, 368 \, 744 \, 177 \, 664\) which contains a zero
\(\displaystyle 2^{47}\) \(=\) \(\displaystyle 140 \, 737 \, 488 \, 355 \, 328\) which contains a zero
\(\displaystyle 2^{48}\) \(=\) \(\displaystyle 281 \, 474 \, 976 \, 710 \, 656\) which contains a zero
\(\displaystyle 2^{49}\) \(=\) \(\displaystyle 562 \, 949 \, 953 \, 421 \, 312\)
\(\displaystyle 2^{50}\) \(=\) \(\displaystyle 1 \, 125 \, 899 \, 906 \, 842 \, 624\) which contains a zero
\(\displaystyle 2^{51}\) \(=\) \(\displaystyle 2 \, 251 \, 799 \, 813 \, 685 \, 248\)
\(\displaystyle 2^{52}\) \(=\) \(\displaystyle 4 \, 503 \, 599 \, 627 \, 370 \, 496\) which contains zeroes
\(\displaystyle 2^{53}\) \(=\) \(\displaystyle 9 \, 007 \, 199 \, 254 \, 740 \, 992\) which contains zeroes
\(\displaystyle 2^{54}\) \(=\) \(\displaystyle 18 \, 014 \, 398 \, 509 \, 481 \, 984\) which contains a zero
\(\displaystyle 2^{55}\) \(=\) \(\displaystyle 36 \, 028 \, 797 \, 018 \, 963 \, 968\) which contains zeroes
\(\displaystyle 2^{56}\) \(=\) \(\displaystyle 72 \, 057 \, 594 \, 037 \, 927 \, 936\) which contains zeroes
\(\displaystyle 2^{57}\) \(=\) \(\displaystyle 144 \, 115 \, 188 \, 075 \, 855 \, 872\) which contains a zero
\(\displaystyle 2^{58}\) \(=\) \(\displaystyle 288 \, 230 \, 376 \, 151 \, 711 \, 744\) which contains a zero
\(\displaystyle 2^{59}\) \(=\) \(\displaystyle 576 \, 460 \, 752 \, 303 \, 423 \, 488\) which contains zeroes
\(\displaystyle 2^{60}\) \(=\) \(\displaystyle 1 \, 152 \, 921 \, 504 \, 606 \, 846 \, 976\) which contains zeroes
\(\displaystyle 2^{61}\) \(=\) \(\displaystyle 2 \, 305 \, 843 \, 009 \, 213 \, 693 \, 952\) which contains zeroes
\(\displaystyle 2^{62}\) \(=\) \(\displaystyle 4 \, 611 \, 686 \, 018 \, 427 \, 387 \, 904\) which contains zeroes
\(\displaystyle 2^{63}\) \(=\) \(\displaystyle 9 \, 223 \, 372 \, 036 \, 854 \, 775 \, 808\) which contains a zero
\(\displaystyle 2^{64}\) \(=\) \(\displaystyle 18 \, 446 \, 744 \, 073 \, 709 \, 551 \, 616\) which contains zeroes
\(\displaystyle 2^{65}\) \(=\) \(\displaystyle 36 \, 893 \, 488 \, 147 \, 419 \, 103 \, 232\) which contains a zero
\(\displaystyle 2^{66}\) \(=\) \(\displaystyle 73 \, 786 \, 976 \, 294 \, 838 \, 206 \, 464\) which contains a zero
\(\displaystyle 2^{67}\) \(=\) \(\displaystyle 147 \, 573 \, 952 \, 589 \, 676 \, 412 \, 928\)
\(\displaystyle 2^{68}\) \(=\) \(\displaystyle 295 \, 147 \, 905 \, 179 \, 352 \, 825 \, 856\) which contains a zero
\(\displaystyle 2^{69}\) \(=\) \(\displaystyle 590 \, 295 \, 810 \, 358 \, 705 \, 651 \, 712\) which contains zeroes
\(\displaystyle 2^{70}\) \(=\) \(\displaystyle 1 \, 180 \, 591 \, 620 \, 717 \, 411 \, 303 \, 424\) which contains zeroes
\(\displaystyle 2^{71}\) \(=\) \(\displaystyle 2 \, 361 \, 183 \, 241 \, 434 \, 822 \, 606 \, 848\) which contains a zero
\(\displaystyle 2^{72}\) \(=\) \(\displaystyle 4 \, 722 \, 366 \, 482 \, 869 \, 645 \, 213 \, 696\)
\(\displaystyle 2^{73}\) \(=\) \(\displaystyle 9 \, 444 \, 732 \, 965 \, 739 \, 290 \, 427 \, 392\) which contains a zero
\(\displaystyle 2^{74}\) \(=\) \(\displaystyle 18 \, 889 \, 465 \, 931 \, 478 \, 580 \, 854 \, 784\) which contains a zero
\(\displaystyle 2^{75}\) \(=\) \(\displaystyle 37 \, 778 \, 931 \, 862 \, 957 \, 161 \, 709 \, 568\) which contains a zero
\(\displaystyle 2^{76}\) \(=\) \(\displaystyle 75 \, 557 \, 863 \, 725 \, 914 \, 323 \, 419 \, 136\)
\(\displaystyle 2^{77}\) \(=\) \(\displaystyle 151 \, 115 \, 727 \, 451 \, 828 \, 646 \, 838 \, 272\)
\(\displaystyle 2^{78}\) \(=\) \(\displaystyle 302 \, 231 \, 454 \, 903 \, 657 \, 293 \, 676 \, 544\) which contains zeroes
\(\displaystyle 2^{79}\) \(=\) \(\displaystyle 604 \, 462 \, 909 \, 807 \, 314 \, 587 \, 353 \, 088\) which contains zeroes
\(\displaystyle 2^{80}\) \(=\) \(\displaystyle 1 \, 208 \, 925 \, 819 \, 614 \, 629 \, 174 \, 706 \, 176\) which contains zeroes
\(\displaystyle 2^{81}\) \(=\) \(\displaystyle 2 \, 417 \, 851 \, 639 \, 229 \, 258 \, 349 \, 412 \, 352\)
\(\displaystyle 2^{82}\) \(=\) \(\displaystyle 4 \, 835 \, 703 \, 278 \, 458 \, 516 \, 698 \, 824 \, 704\) which contains zeroes
\(\displaystyle 2^{83}\) \(=\) \(\displaystyle 9 \, 671 \, 406 \, 556 \, 917 \, 033 \, 397 \, 649 \, 408\) which contains zeroes
\(\displaystyle 2^{84}\) \(=\) \(\displaystyle 19 \, 342 \, 813 \, 113 \, 834 \, 066 \, 795 \, 298 \, 816\) which contains a zero
\(\displaystyle 2^{85}\) \(=\) \(\displaystyle 38 \, 685 \, 626 \, 227 \, 668 \, 133 \, 590 \, 597 \, 632\) which contains a zero
\(\displaystyle 2^{86}\) \(=\) \(\displaystyle 77 \, 371 \, 252 \, 455 \, 336 \, 267 \, 181 \, 195 \, 264\)

$\blacksquare$


Sources