40144044691：極端に弱い素数

$40144044691$ は素数です。

みんな大好き $691$ で終わり、 $4$ がたくさん出てくるので覚えやすいですが、面白い特徴があります。

まず、 $40144044691$ は弱い素数です。つまり、どの桁であっても一つ数を変更すると素数でなくなります。

$\begin{align} (\color{red}{0}0)144044691 &= 3 \times 7 \times 6859271\\ \color{red}{1}0144044691 &=11 \times 19 \times 79 \times 113 \times 5437\\ \color{red}{2}0144044691 &=277 \times 569 \times 127807 \\ \color{red}{3}0144044691 &=3^2 \times 47 \times 71 262517 \\ \color{red}{5}0144044691 &=13 \times 3857 234207 \\ \color{red}{6}0144044691 &=3 \times 71 \times 101 \times 139 \times 20113 \\ \color{red}{7}0144044691 &=7 \times 10020 577813 \\ \color{red}{8}0144044691 &=33827 \times 2 369233\\ \color{red}{9}0144044691 &=3 \times 421 \times 601 \times 118757 \\ 4\color{red}{1}144044691 &=17 \times 269 \times 8 997167 \\ 4\color{red}{2}144044691 &=3 \times 7 \times 877 \times 2 288323 \\ 4\color{red}{3}144044691 &=11 \times 29 \times 167 \times 197 \times 4111 \\ 4\color{red}{4}144044691 &=4391 \times 10 053301 \\ 4\color{red}{5}144044691 &=3 \times 1741 \times 8 643317 \\ 4\color{red}{6}144044691 &=37 \times 1247 136343 \\ 4\color{red}{7}144044691 &=160409 \times 293899 \\ 4\color{red}{8}144044691 &=3^2 \times 19 \times 281 544121 \\ 4\color{red}{9}144044691 &=7 \times 7020 577813 \\ 40\color{red}{0}44044691 &=3^2 \times 7 \times 97 \times 6 552781 \\ 40\color{red}{2}44044691 &=29 \times 1387 725679 \\ 40\color{red}{3}44044691 &=3 \times 1613 \times 8 337269 \\ 40\color{red}{4}44044691 &=16447 \times 2 459053 \\ 40\color{red}{5}44044691 &=19 \times 271 \times 331 \times 23789 \\ 40\color{red}{6}44044691 &=3 \times 6823 \times 1 985639 \\ 40\color{red}{7}44044691 &=7 \times 733 \times 7 940761 \\ 40\color{red}{8}44044691 &=1979 \times 20 638729 \\ 40\color{red}{9}44044691 &=3^2 \times 11 \times 103 \times 4 015303 \\ 401\color{red}{0}4044691 &=3 \times 23 \times 31 \times 409 \times 45841 \\ 401\color{red}{1}4044691 &=7 \times 1019 \times 5 623727 \\ 401\color{red}{2}4044691 &=17 \times 53 \times 44 532791 \\ 401\color{red}{3}4044691 &=3^2 \times 13 \times 343 026023 \\ 401\color{red}{5}4044691 &=2243 \times 17 901937 \\ 401\color{red}{6}4044691 &=3 \times 19 \times 79 \times 131 \times 68087 \\ 401\color{red}{7}4044691 &=11 \times 523 \times 6 983147 \\ 401\color{red}{8}4044691 &=7 \times 41 \times 61 \times 2 295313 \\ 401\color{red}{9}4044691 &=3 \times 947 \times 2801 \times 5051 \\ 4014\color{red}{0}044691 &=3 \times 13380 014897 \\ 4014\color{red}{1}044691 &=11^3 \times 17 \times 97 \times 18289 \\ 4014\color{red}{2}044691 &=7 \times 563 \times 10 185751 \\ 4014\color{red}{3}044691 &=3^3 \times 41 \times 36 262913 \\ 4014\color{red}{5}044691 &=19 \times 2112 897089 \\ 4014\color{red}{6}044691 &=3 \times 101 \times 617 \times 214741 \\ 4014\color{red}{7}044691 &=13 \times 23497 \times 131431 \\ 4014\color{red}{8}044691 &=227 \times 11243 \times 15731 \\ 4014\color{red}{9}044691 &=3 \times 7^2 \times 67 \times 4 076459 \\ 40144\color{red}{1}44691 &=72 \times 819 268259 \\ 40144\color{red}{2}44691 &=3 \times 13381 414897 \\ 40144\color{red}{3}44691 &=11 \times 31 \times 157 \times 749843 \\ 40144\color{red}{4}44691 &=13 \times 17 \times 43 \times 4 224397 \\ 40144\color{red}{5}44691 &=3 \times 8527 \times 1 569311 \\ 40144\color{red}{6}44691 &=1229 \times 32 664479 \\ 40144\color{red}{7}44691 &=12613 \times 3 182807 \\ 40144\color{red}{8}44691 &=3^2 \times 7 \times 16427 \times 38791 \\ 40144\color{red}{9}44691 &=149 \times 409 \times 658751 \\ 401440\color{red}{0}4691 &=3 \times 7 \times 661 \times 709 \times 4079 \\ 401440\color{red}{1}4691 &=11 \times 43 \times 1201 \times 70667 \\ 401440\color{red}{2}4691 &=101 \times 463 \times 858457 \\ 401440\color{red}{3}4691 &=3^2 \times 31 \times 143 885429 \\ 401440\color{red}{5}4691 &=13 \times 71 \times 43 493017 \\ 401440\color{red}{6}4691 &=3 \times 23 \times 18541 \times 31379 \\ 401440\color{red}{7}4691 &=7 \times 89 \times 137 \times 193 \times 2437 \\ 401440\color{red}{8}4691 &=223 \times 2389 \times 75353 \\ 401440\color{red}{9}4691 &=3 \times 19 \times 704 282363 \\ 4014404\color{red}{0}691 &=3 \times 13381 346897 \\ 4014404\color{red}{1}691 &=13 \times 23 \times 134 261009 \\ 4014404\color{red}{2}691 &=6359 \times 6 312949 \\ 4014404\color{red}{3}691 &=3^3 \times 83 \times 103 \times 173917 \\ 4014404\color{red}{5}691 &=263 \times 397 \times 384481 \\ 4014404\color{red}{6}691 &=3 \times 7^3 \times 47 \times 251 \times 3307 \\ 4014404\color{red}{7}691 &=11 \times 41641 \times 87641 \\ 4014404\color{red}{8}691 &=271 \times 3779 \times 39199 \\ 4014404\color{red}{9}691 &=3 \times 1151 \times 1327 \times 8761 \\ 40144044\color{red}{0}91 &=6971 \times 5 758721 \\ 40144044\color{red}{1}91 &=159491 \times 251701 \\ 40144044\color{red}{2}91 &=3 \times 13 \times 1029 334469 \\ 40144044\color{red}{3}91 &=11 \times 24281 \times 150301 \\ 40144044\color{red}{4}91 &=31337 \times 1 281043 \\ 40144044\color{red}{5}91 &=3^2 \times 7 \times 637 207057 \\ 40144044\color{red}{7}91 &=43 \times 933 582437 \\ 40144044\color{red}{8}91 &=3 \times 101 \times 409 \times 323933 \\ 40144044\color{red}{9}91 &=193003 \times 207997 \\ 401440446\color{red}{0}1 &=126001 \times 318601 \\ 401440446\color{red}{1}1 &=11 \times 31 \times 117 724471 \\ 401440446\color{red}{2}1 &=3 \times 233 \times 57 430679 \\ 401440446\color{red}{3}1 &=823 \times 48 777697 \\ 401440446\color{red}{4}1 &=5279 \times 7 604479 \\ 401440446\color{red}{5}1 &=3 \times 78607 \times 170231 \\ 401440446\color{red}{6}1 &=7 \times 5734 863523 \\ 401440446\color{red}{7}1 &=2543 \times 15 786097 \\ 401440446\color{red}{8}1 &=3^2 \times 13 \times 17^2 \times 23 \times 41 \times 1259 \\ 401440446\color{red}{9}0 &=2 \times 3^5 \times 5 \times 16 520183 \\ 4014404469\color{red}{2} &=2^2 \times 109 \times 2719 \times 33863 \\ 4014404469\color{red}{3} &=3 \times 1321 \times 10 129711 \\ 4014404469\color{red}{4} &=2 \times 13^2 \times 613 \times 193751 \\ 4014404469\color{red}{5} &=5 \times 8028 808939 \\ 4014404469\color{red}{6} &=2^3 \times 3 \times 7 \times 211 \times 1 132477 \\ 4014404469\color{red}{7} &=37 \times 1084 974181 \\ 4014404469\color{red}{8} &=2 \times 17 \times 89 \times 3137 \times 4229 \\ 4014404469\color{red}{9} &=3^2 \times 11 \times 405 495401 \end{align}$

実は、この $40144044691$ はただの弱い素数ではありません。

例えば、最小の弱い素数は $294001$ でしたが、一万の位を消去した $2\color{red}{|}4001$ は素数になります。

しかし、 $40144044691$ はどの桁の数を一つ取り除いても素数ではありません。

$\begin{align} \color{red}{|}(0)144044691 &=3 \times 7 \times 6 859271 \\ 4\color{red}{|}144044691 &=31 \times 3307 \times 40423 \\ 40\color{red}{|}44044691 &=3^3 \times 3881 \times 38593 \\ 401\color{red}{|}4044691 &=3 \times 1523 \times 878539 \\ 4014\color{red}{|}044691 &=3 \times 1523 \times 878539 \\ 40144\color{red}{|}44691 &=1091 \times 3 679601 \\ 401440\color{red}{|}4691 &=3 \times 11 \times 121 648627 \\ 4014404\color{red}{|}691 &=3 \times 11 \times 121 648627 \\ 40144044\color{red}{|}91 &=109 \times 36 829399 \\ 401440446\color{red}{|}1 &=97 \times 5653 \times 7321 \\ 4014404469\color{red}{|} &=3^5 \times 16 520183 \end{align}$

また、 $294001$ に一つ数を挿入して $2\color{red}{6}94001$ のように素数にすることができます。

ですが、 $40144044691$ はどこに $0$ から $9$ の数を一つ挿入しても素数ではありません！*1

$\begin{align}\color{red}{1}40144044691 &=7 \times 20020 577813 \\ \color{red}{2}40144044691 &=3 \times 67 \times 197 \times 6 064703 \\ \color{red}{3}40144044691 &=11 \times 30922 185881 \\ \color{red}{4}40144044691 &=13 \times 76829 \times 440683 \\ \color{red}{5}40144044691 &=3 \times 19267 \times 9 344891 \\ \color{red}{6}40144044691 &=131 \times 191 \times 25 584271 \\ \color{red}{7}40144044691 &=211 \times 3507 791681 \\ \color{red}{8}40144044691 &=3^2 \times 7 \times 17^2 \times 5827 \times 7919 \\ \color{red}{9}40144044691 &=2089 \times 10463 \times 43013 \\ 4\color{red}{0}0144044691 &=19081 \times 20 970811 \\ 4\color{red}{1}0144044691 &=107 \times 239 \times 337 \times 47591 \\ 4\color{red}{2}0144044691 &=3 \times 7^2 \times 29 \times 43 \times 2 291999 \\ 4\color{red}{3}0144044691 &=118147 \times 3 640753 \\ 4\color{red}{4}0144044691 &=13 \times 76829 \times 440683 \\ 4\color{red}{5}0144044691 &=3 \times 11 \times 26723 \times 510449 \\ 4\color{red}{6}0144044691 &=375511 \times 1 225381 \\ 4\color{red}{7}0144044691 &=149 \times 283 \times 2129 \times 5237 \\ 4\color{red}{8}0144044691 &=3^2 \times 53349 338299 \\ 4\color{red}{9}0144044691 &=7 \times 37 \times 479 \times 1279 \times 3089 \\ 40\color{red}{0}144044691 &=19081 \times 20 970811 \\ 40\color{red}{1}144044691 &=13 \times 30857 234207 \\ 40\color{red}{2}144044691 &=3 \times 21529 \times 6 226393 \\ 40\color{red}{3}144044691 &=157 \times 2567 796463 \\ 40\color{red}{4}144044691 &=307 \times 4273 \times 308081 \\ 40\color{red}{5}144044691 &=3 \times 79 \times 1709 468543 \\ 40\color{red}{6}144044691 &=7 \times 11 \times 47 \times 112 225489 \\ 40\color{red}{7}144044691 &=31 \times 13133 678861 \\ 40\color{red}{8}144044691 &=3^2 \times 991 \times 4651 \times 9839 \\ 40\color{red}{9}144044691 &=19 \times 1321 \times 16 301209 \\ 401\color{red}{0}44044691 &=523 \times 766 814617 \\ 401\color{red}{1}44044691 &=13 \times 30857 234207 \\ 401\color{red}{2}44044691 &=3 \times 7 \times 86539 \times 220789 \\ 401\color{red}{3}44044691 &=37 \times 10847 136343 \\ 401\color{red}{4}44044691 &=47 \times 103 \times 82 925851 \\ 401\color{red}{5}44044691 &=3 \times 17 \times 19 \times 414 390139 \\ 401\color{red}{6}44044691 &=127 \times 3162 551533 \\ 401\color{red}{7}44044691 &=11 \times 643 \times 2237 \times 25391 \\ 401\color{red}{8}44044691 &=3^2 \times 44649 338299 \\ 401\color{red}{9}44044691 &=7 \times 311 \times 9221 \times 20023 \\ 4014\color{red}{0}4044691 &=13 \times 30877 234207 \\ 4014\color{red}{1}4044691 &=11 \times 36492 185881 \\ 4014\color{red}{2}4044691 &=3 \times 133808 014897 \\ 4014\color{red}{3}4044691 &=23 \times 971 \times 1831 \times 9817 \\ 4014\color{red}{4}4044691 &=47 \times 103 \times 82 925851 \\ 4014\color{red}{5}4044691 &=3 \times 7 \times 44519 \times 429409 \\ 4014\color{red}{6}4044691 &=179833 \times 2 232427 \\ 4014\color{red}{7}4044691 &=15173 \times 26 459767 \\ 4014\color{red}{8}4044691 &=3^3 \times 14869 779433 \\ 4014\color{red}{9}4044691 &=10399 \times 38 608909 \\ 40144\color{red}{0}044691 &=7 \times 31 \times 10567 \times 175069 \\ 40144\color{red}{1}044691 &=71 \times 5654 099221 \\ 40144\color{red}{2}044691 &=3 \times 17^2 \times 3967 \times 116719 \\ 40144\color{red}{3}044691 &=13 \times 1237 \times 24 963811 \\ 40144\color{red}{4}044691 &=47 \times 103 \times 82 925851 \\ 40144\color{red}{5}044691 &=3 \times 133815 014897 \\ 40144\color{red}{6}044691 &=59 \times 281 \times 2609 \times 9281 \\ 40144\color{red}{7}044691 &=7 \times 11^2 \times 1013 \times 467881 \\ 40144\color{red}{8}044691 &=3^2 \times 673 \times 66 278363 \\ 40144\color{red}{9}044691 &=19 \times 21128 897089 \\ 401440\color{red}{0}44691 &=7 \times 31 \times 10567 \times 175069 \\ 401440\color{red}{1}44691 &=59 \times 149 \times 953 \times 47917 \\ 401440\color{red}{2}44691 &=3 \times 37 \times 13513 \times 267637 \\ 401440\color{red}{3}44691 &=17 \times 23614 137923 \\ 401440\color{red}{4}44691 &=11 \times 13 \times 113 \times 24 843149 \\ 401440\color{red}{5}44691 &=3 \times 229 \times 584 338493 \\ 401440\color{red}{6}44691 &=397 \times 4871 \times 207593 \\ 401440\color{red}{7}44691 &=7^2 \times 10477 \times 781967 \\ 401440\color{red}{8}44691 &=3^5 \times 127 \times 1201 \times 10831 \\ 401440\color{red}{9}44691 &=23 \times 53 \times 329 319889 \\ 4014404\color{red}{0}4691 &=173 \times 2320 464767 \\ 4014404\color{red}{1}4691 &=53 \times 401 \times 499 \times 37853 \\ 4014404\color{red}{2}4691 &=3 \times 281 \times 476 204537 \\ 4014404\color{red}{3}4691 &=71761 \times 5 594131 \\ 4014404\color{red}{4}4691 &=11 \times 13 \times 113 \times 24 843149 \\ 4014404\color{red}{5}4691 &=3 \times 1181 \times 113 305237 \\ 4014404\color{red}{6}4691 &=7 \times 89 \times 863 \times 746659 \\ 4014404\color{red}{7}4691 &=137 \times 139 \times 251 \times 83987 \\ 4014404\color{red}{8}4691 &=3^2 \times 23 \times 29201 \times 66413 \\ 4014404\color{red}{9}4691 &=19^2 \times 29 \times 431 \times 88969 \\ 40144044\color{red}{0}691 &=61 \times 1373 \times 4 793147 \\ 40144044\color{red}{1}691 &=6317 \times 63 549223 \\ 40144044\color{red}{2}691 &=3 \times 149 \times 4003 \times 224351 \\ 40144044\color{red}{3}691 &=7 \times 59417 \times 965189 \\ 40144044\color{red}{4}691 &=11 \times 13 \times 113 \times 24 843149 \\ 40144044\color{red}{5}691 &=3 \times 79 \times 30763 \times 55061 \\ 40144044\color{red}{6}691 &=17 \times 23614 143923 \\ 40144044\color{red}{7}691 &=31 \times 67 \times 1783 \times 108401 \\ 40144044\color{red}{8}691 &=3^2 \times 44604 494299 \\ 40144044\color{red}{9}691 &=3061 \times 131 146831 \\ 401440446\color{red}{0}91 &=2963 \times 135 484457 \\ 401440446\color{red}{1}91 &=397 \times 1011 185003 \\ 401440446\color{red}{2}91 &=3 \times 133813 482097 \\ 401440446\color{red}{3}91 &=97 \times 4138 561303 \\ 401440446\color{red}{4}91 &=7 \times 43 \times 53 \times 349 \times 72103 \\ 401440446\color{red}{5}91 &=3 \times 3671 \times 36 451507 \\ 401440446\color{red}{6}91 &=17 \times 23614 143923 \\ 401440446\color{red}{7}91 &=61 \times 337 \times 4261 \times 4583 \\ 401440446\color{red}{8}91 &=3^2 \times 11 \times 89 \times 139 \times 327779 \\ 401440446\color{red}{9}91 &=101 \times 569 \times 1237 \times 5647 \\ 4014404469\color{red}{0}1 &=13 \times 103 \times 299 806159 \\ 4014404469\color{red}{1}1 &=7 \times 577 \times 99 391049 \\ 4014404469\color{red}{2}1 &=3 \times 43 \times 13291 \times 234139 \\ 4014404469\color{red}{3}1 &=373 \times 3643 \times 295429 \\ 4014404469\color{red}{4}1 &=4139 \times 96 989719 \\ 4014404469\color{red}{5}1 &=3 \times 113 \times 1184 190109 \\ 4014404469\color{red}{6}1 &=269 \times 16427 \times 90847 \\ 4014404469\color{red}{7}1 &=23 \times 619 \times 3541 \times 7963 \\ 4014404469\color{red}{8}1 &=3^4 \times 7 \times 708 007843 \\ 4014404469\color{red}{9}1 &=101 \times 569 \times 1237 \times 5647 \\ 40144044691\color{red}{0} &=2 \times 5 \times 40144 044691 \\ 40144044691\color{red}{1} &=7 \times 577 \times 99 391049 \\ 40144044691\color{red}{2} &=2^6 \times 3 \times 17 \times 347 \times 354439 \\ 40144044691\color{red}{3} &=11 \times 61 \times 598 271903 \\ 40144044691\color{red}{4} &=2 \times 13 \times 15440 017189 \\ 40144044691\color{red}{5} &=3 \times 5 \times 53 \times 3433 \times 147089 \\ 40144044691\color{red}{6} &=2^2 \times 31 \times 263 \times 12 309593 \\ 40144044691\color{red}{7} &=18787 \times 21 367991 \\ 40144044691\color{red}{8} &=2 \times 3^2 \times 7^2 \times 1063 \times 428173 \\ 40144044691\color{red}{9} &=179 \times 9613 \times 233297 \\ \end{align}$