Powers of 5 with no Zero in Decimal Representation

Unproven Hypotheses

The following powers of $5$ which contain no zero in their decimal representation are believed to be all that exist:

$1, 5, 25, 125, 625, 3125, 15 \, 625, 78 \, 125, 1 \, 953 \, 125, 9 \, 765 \, 625,$
$48 \, 828 \, 125, 762 \, 939 \, 453 \, 125, 3 \, 814 \, 697 \, 265 \, 625, 931 \, 322 \, 574 \, 615 \, 478 \, 515 \, 625,$
$116 \, 415 \, 321 \, 826 \, 934 \, 814 \, 453 \, 125, 34 \, 694 \, 469 \, 519 \, 536 \, 141 \, 888 \, 238 \, 489 \, 627 \, 838 \, 134 \, 765 \, 625$

but this has not been proven.

The corresponding indices are:

$0, 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 17, 18, 30, 33, 58$

Progress

 $\displaystyle 5^0$ $=$ $\displaystyle 1$ $\displaystyle 5^1$ $=$ $\displaystyle 2$ $\displaystyle 5^2$ $=$ $\displaystyle 25$ $\displaystyle 5^3$ $=$ $\displaystyle 125$ $\displaystyle 5^4$ $=$ $\displaystyle 625$ $\displaystyle 5^5$ $=$ $\displaystyle 3125$ $\displaystyle 5^6$ $=$ $\displaystyle 15 \, 625$ $\displaystyle 5^7$ $=$ $\displaystyle 78 \, 125$ $\displaystyle 5^8$ $=$ $\displaystyle 390 \, 625$ which contains a zero $\displaystyle 5^9$ $=$ $\displaystyle 1 \, 953 \, 125$ $\displaystyle 5^{10}$ $=$ $\displaystyle 9 \, 765 \, 625$ $\displaystyle 5^{11}$ $=$ $\displaystyle 48 \, 828 \, 125$ $\displaystyle 5^{12}$ $=$ $\displaystyle 244 \, 140 \, 625$ which contains a zero $\displaystyle 5^{13}$ $=$ $\displaystyle 1 \, 220 \, 703 \, 125$ which contains zeroes $\displaystyle 5^{14}$ $=$ $\displaystyle 6 \, 103 \, 515 \, 625$ which contains a zero $\displaystyle 5^{15}$ $=$ $\displaystyle 30 \, 517 \, 578 \, 125$ which contains a zero $\displaystyle 5^{16}$ $=$ $\displaystyle 152 \, 587 \, 890 \, 625$ which contains a zero $\displaystyle 5^{17}$ $=$ $\displaystyle 762 \, 939 \, 453 \, 125$ $\displaystyle 5^{18}$ $=$ $\displaystyle 3 \, 814 \, 697 \, 265 \, 625$ $\displaystyle 5^{19}$ $=$ $\displaystyle 19 \, 073 \, 486 \, 328 \, 125$ which contains a zero $\displaystyle 5^{20}$ $=$ $\displaystyle 95 \, 367 \, 431 \, 640 \, 625$ which contains a zero $\displaystyle 5^{21}$ $=$ $\displaystyle 476 \, 837 \, 158 \, 203 \, 125$ which contains a zero $\displaystyle 5^{22}$ $=$ $\displaystyle 2 \, 384 \, 185 \, 791 \, 015 \, 625$ which contains a zero $\displaystyle 5^{23}$ $=$ $\displaystyle 11 \, 920 \, 928 \, 955 \, 078 \, 125$ which contains zeroes $\displaystyle 5^{24}$ $=$ $\displaystyle 59 \, 604 \, 644 \, 775 \, 390 \, 625$ which contains zeroes $\displaystyle 5^{25}$ $=$ $\displaystyle 298 \, 023 \, 223 \, 876 \, 953 \, 125$ which contains a zero $\displaystyle 5^{26}$ $=$ $\displaystyle 1 \, 490 \, 116 \, 119 \, 384 \, 765 \, 625$ which contains a zero $\displaystyle 5^{27}$ $=$ $\displaystyle 7 \, 450 \, 580 \, 596 \, 923 \, 828 \, 125$ which contains zeroes $\displaystyle 5^{28}$ $=$ $\displaystyle 37 \, 252 \, 902 \, 984 \, 619 \, 140 \, 625$ which contains zeroes $\displaystyle 5^{29}$ $=$ $\displaystyle 186 \, 264 \, 514 \, 923 \, 095 \, 703 \, 125$ which contains zeroes $\displaystyle 5^{30}$ $=$ $\displaystyle 931 \, 322 \, 574 \, 615 \, 478 \, 515 \, 625$ $\displaystyle 5^{31}$ $=$ $\displaystyle 4 \, 656 \, 612 \, 873 \, 077 \, 392 \, 578 \, 125$ which contains a zero $\displaystyle 5^{32}$ $=$ $\displaystyle 23 \, 283 \, 064 \, 365 \, 386 \, 962 \, 890 \, 625$ which contains zeroes $\displaystyle 5^{33}$ $=$ $\displaystyle 116 \, 415 \, 321 \, 826 \, 934 \, 814 \, 453 \, 125$ $\displaystyle 5^{34}$ $=$ $\displaystyle 582 \, 076 \, 609 \, 134 \, 674 \, 072 \, 265 \, 625$ which contains zeroes $\displaystyle 5^{35}$ $=$ $\displaystyle 2 \, 910 \, 383 \, 045 \, 673 \, 370 \, 361 \, 328 \, 125$ which contains zeroes $\displaystyle 5^{36}$ $=$ $\displaystyle 14 \, 551 \, 915 \, 228 \, 366 \, 851 \, 806 \, 640 \, 625$ which contains zeroes $\displaystyle 5^{37}$ $=$ $\displaystyle 72 \, 759 \, 576 \, 141 \, 834 \, 259 \, 033 \, 203 \, 125$ which contains zeroes $\displaystyle 5^{38}$ $=$ $\displaystyle 363 \, 797 \, 880 \, 709 \, 171 \, 295 \, 166 \, 015 \, 625$ which contains zeroes $\displaystyle 5^{39}$ $=$ $\displaystyle 1 \, 818 \, 989 \, 403 \, 545 \, 856 \, 475 \, 830 \, 078 \, 125$ which contains zeroes $\displaystyle 5^{40}$ $=$ $\displaystyle 9 \, 094 \, 947 \, 017 \, 729 \, 282 \, 379 \, 150 \, 390 \, 625$ which contains zeroes $\displaystyle 5^{41}$ $=$ $\displaystyle 45 \, 474 \, 735 \, 088 \, 646 \, 411 \, 895 \, 751 \, 953 \, 125$ which contains a zero $\displaystyle 5^{42}$ $=$ $\displaystyle 227 \, 373 \, 675 \, 443 \, 232 \, 059 \, 478 \, 759 \, 765 \, 625$ which contains a zero $\displaystyle 5^{43}$ $=$ $\displaystyle 1 \, 136 \, 868 \, 377 \, 216 \, 160 \, 297 \, 393 \, 798 \, 828 \, 125$ which contains a zero $\displaystyle 5^{44}$ $=$ $\displaystyle 5 \, 684 \, 341 \, 886 \, 080 \, 801 \, 486 \, 968 \, 994 \, 140 \, 625$ which contains zeroes $\displaystyle 5^{45}$ $=$ $\displaystyle 28 \, 421 \, 709 \, 430 \, 404 \, 007 \, 434 \, 844 \, 970 \, 703 \, 125$ which contains zeroes $\displaystyle 5^{46}$ $=$ $\displaystyle 142 \, 108 \, 547 \, 152 \, 020 \, 037 \, 174 \, 224 \, 853 \, 515 \, 625$ which contains zeroes $\displaystyle 5^{47}$ $=$ $\displaystyle 710 \, 542 \, 735 \, 760 \, 100 \, 185 \, 871 \, 124 \, 267 \, 578 \, 125$ which contains zeroes $\displaystyle 5^{48}$ $=$ $\displaystyle 3 \, 552 \, 713 \, 678 \, 800 \, 500 \, 929 \, 355 \, 621 \, 337 \, 890 \, 625$ which contains zeroes $\displaystyle 5^{49}$ $=$ $\displaystyle 17 \, 763 \, 568 \, 394 \, 002 \, 504 \, 646 \, 778 \, 106 \, 689 \, 453 \, 125$ which contains zeroes $\displaystyle 5^{50}$ $=$ $\displaystyle 88 \, 817 \, 841 \, 970 \, 012 \, 523 \, 233 \, 890 \, 533 \, 447 \, 265 \, 625$ which contains zeroes $\displaystyle 5^{51}$ $=$ $\displaystyle 444 \, 089 \, 209 \, 850 \, 062 \, 616 \, 169 \, 452 \, 667 \, 236 \, 328 \, 125$ which contains zeroes $\displaystyle 5^{52}$ $=$ $\displaystyle 2 \, 220 \, 446 \, 049 \, 250 \, 313 \, 080 \, 847 \, 263 \, 336 \, 181 \, 640 \, 625$ which contains zeroes $\displaystyle 5^{53}$ $=$ $\displaystyle 11 \, 102 \, 230 \, 246 \, 251 \, 565 \, 404 \, 236 \, 316 \, 680 \, 908 \, 203 \, 125$ which contains zeroes $\displaystyle 5^{54}$ $=$ $\displaystyle 55 \, 511 \, 151 \, 231 \, 257 \, 827 \, 021 \, 181 \, 583 \, 404 \, 541 \, 015 \, 625$ which contains zeroes $\displaystyle 5^{55}$ $=$ $\displaystyle 277 \, 555 \, 756 \, 156 \, 289 \, 135 \, 105 \, 907 \, 917 \, 022 \, 705 \, 078 \, 125$ which contains zeroes $\displaystyle 5^{56}$ $=$ $\displaystyle 1 \, 387 \, 778 \, 780 \, 781 \, 445 \, 675 \, 529 \, 539 \, 585 \, 113 \, 525 \, 390 \, 625$ which contains zeroes $\displaystyle 5^{57}$ $=$ $\displaystyle 6 \, 938 \, 893 \, 903 \, 907 \, 228 \, 377 \, 647 \, 697 \, 925 \, 567 \, 626 \, 953 \, 125$ which contains zeroes $\displaystyle 5^{58}$ $=$ $\displaystyle 34 \, 694 \, 469 \, 519 \, 536 \, 141 \, 888 \, 238 \, 489 \, 627 \, 838 \, 134 \, 765 \, 625$

$\blacksquare$