{"id":141,"date":"2026-08-20T19:26:19","date_gmt":"2026-08-20T19:26:19","guid":{"rendered":"https:\/\/mathrelated.co.uk\/?p=141"},"modified":"2026-09-03T18:38:21","modified_gmt":"2026-09-03T18:38:21","slug":"super-element-13t-and-set-1-13t3t1","status":"publish","type":"post","link":"https:\/\/mathrelated.co.uk\/index.php\/2026\/08\/20\/super-element-13t-and-set-1-13t3t1\/","title":{"rendered":"Element -1+3^t and set 1+(-1+3^t)*(3^(t+1))"},"content":{"rendered":"\n

Consider the Proth space <\/p>\n\n\n\n

1<\/mn>+<\/mo>(<\/mo>\u2212<\/mo>1<\/mn>+<\/mo>3<\/mn>t<\/mi><\/msup>)<\/mo>\u2217<\/mo>(<\/mo>3<\/mn>t<\/mi>+<\/mo>1<\/mn><\/mrow><\/msup>)<\/mo><\/mrow>1+(-1+3^t)*(3^{t+1})<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

We will be using generating element \u22121+3^t
<\/p>\n\n\n\n

The [sub]groups generated by \u22121+3^t have low order [ a lot lower than n-1 ]<\/p>\n\n\n\n

Tabulating the elements of groups generated by \u22121+3^t next.
<\/p>\n\n\n\n

Using the ENt to avoid clashing with existing letter conventions for groups. Prefer G or S? Replace them in your local copy.<\/p>\n\n\n\n

For 8 from t=2 elements ENt5<\/strong> are { 8, 64, 78, 190, 1 }<\/p>\n\n\n\n

For 26 from t= 3 the elements in ENt42<\/strong> are as shown next.
{ 26, 676, 720, 1864, 3, 78, 2028, 53, 1378, 9
234, 1870, 159, 2027, 27, 702, 1396, 477, 1867, 81
2106, 2081, 1431, 1387, 243, 2104, 2029, 79, 2054, 729
2098, 1873, 237, 1948, 80, 2080, 1405, 711, 1630, 240
2026, 1 }<\/p>\n\n\n\n

When t=4 we have P=1+80\u00d7243 and the set of elements modulo P is a multiplicative group.
<\/p>\n\n\n\n

Our generator \u22121+3^t is 80 and the subset it generates has 27 elements.
<\/p>\n\n\n\n

The elements in ENt27<\/strong> are as shown next
{ 80, 6400, 6534, 17254, 9, 720, 18718, 483, 19199, 81,
6480, 12934, 4347, 17263, 729, 19438, 19201, 241, 19280, 6561,
19414, 17281, 2169, 17992, 726, 19198, 1 }<\/p>\n\n\n\n

If ENt27<\/strong> is really a group then we need inverses so let us document those next.<\/p>\n\n\n\n

    \n
  • Inverse of 80 is 19198 mod P<\/li>\n\n\n\n
  • Inverse of 6400 is 726 mod P<\/li>\n\n\n\n
  • Inverse of 6534 is 17992 mod P<\/li>\n\n\n\n
  • Inverse of 17254 is 2169 mod P<\/li>\n\n\n\n
  • Inverse of 9 is 17281 mod P<\/li>\n\n\n\n
  • Inverse of 720 is 19414 mod P<\/li>\n\n\n\n
  • Inverse of 18718 is 6561 mod P<\/li>\n\n\n\n
  • Inverse of 483 is 19280 mod P<\/li>\n\n\n\n
  • Inverse of 19199 is 241 mod P<\/li>\n\n\n\n
  • Inverse of 81 is 19201 mod P<\/li>\n\n\n\n
  • Inverse of 6480 is 19438 mod P<\/li>\n\n\n\n
  • Inverse of 12934 is 729 mod P<\/li>\n\n\n\n
  • Inverse of 4347 is 17263 mod P<\/li>\n<\/ul>\n\n\n\n

    For 242 from t=5 the 22 elements in ENt22<\/strong> are as shown next
    { 242, 58564, 58968, 156736, 27, 6534, 169876, 4365,
    174235, 729, 176418, 176177, 117855, 117451, 19683, 176392,
    169885, 6543, 172054, 2184, 175690, 1 }<\/p>\n\n\n\n

    The seventh to last element is always -27<\/p>\n\n\n\n

    The thirteenth to last element is always 729 alternatively written as 3^6<\/p>\n\n\n\n

    For 728 from t=6 the 39 elements in ENt39<\/strong> are as shown next
    { 728, 529984, 531198<\/strong>, 1414990, 81, 58968,
    1533142, 39339, 1572463, 6561, 1592134, 1589953,
    2185, 1590680, 531441<\/strong>, 1591894, 1415233, 176985,
    1474120, 59022, 1572454, 9, 6552, 1585582,
    4371, 1589951, 729<\/strong>, 530712, 1061182, 354051, 1415071, 59049,
    1592110, 1572481, 19665, 1579024, 6558, 1589950, 1 }<\/p>\n\n\n\n

    Every twelfth element after 3^6 from the end of the subgroup is a power of 3.<\/p>\n\n\n\n

    In order we would write them as follows<\/p>\n\n\n\n

    3<\/mn>6<\/mn><\/msup>,<\/mo>3<\/mn>12<\/mn><\/msup>,<\/mo>3<\/mn>18<\/mn><\/msup>,<\/mo>3<\/mn>24<\/mn><\/msup>,<\/mo>3<\/mn>30<\/mn><\/msup>,<\/mo>3<\/mn>36<\/mn><\/msup>,<\/mo>3<\/mn>42<\/mn><\/msup>,<\/mo>3<\/mn>48<\/mn><\/msup>,<\/mo>3<\/mn>54<\/mn><\/msup>,<\/mo>3<\/mn>60<\/mn><\/msup><\/mrow>3^6, 3^{12}, 3^{18}, 3^{24}, 3^{30}, 3^{36}, 3^{42}, 3^{48}, 3^{54}, 3^{60}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

    Writing them longhand we might say
    <\/p>\n\n\n\n

      \n
    • thirteenth from last is 3^6<\/li>\n\n\n\n
    • twentyfifth from last is 3^12<\/li>\n\n\n\n
    • \u2026<\/li>\n\n\n\n
    • fortyninth from last is 3^24<\/li>\n\n\n\n
    • …<\/li>\n<\/ul>\n\n\n\n

      In calculation terms we can jump to any portion of our group and be at most 11 elements away from the element we require.
      <\/p>\n\n\n\n

      This can be a tremendous labour saver as instead of repeated powering to obtain an element of the subgroup, we can jump near our required element and then conduct a much smaller number of repeated powering to gain our result.<\/p>\n\n\n\n

      For 2186 from t=7 the 90 elements in ENt90<\/strong> are as shown next.
      { 2186,4778596,4782240,12748024,243,531198<\/strong>,
      13811068,354213,14165227,59049,14342338,14322673,
      19677,14329228,6560,14340160,9561565,4781511<\/strong>,
      11154430,1594080,13810906,81,177066,14165254,
      118071,14283307,19683,14342344,14335789,6559<\/strong>,
      14337974,4782969,14341618,12748753,1593837,13279708
      531360,14165200,27,59022,14283316,39357<\/strong>,
      14322667,6561,-1<\/strong>,14340161,9563751,9560107,
      1594323,14342104,13811149,531279,13988134,177120<\/strong>,
      14283298,9,19674,14322670,13119,14335787,
      2187,4780782,9560836,3187917,12748267,531441<\/strong>,
      14342266,14165281,177093,14224276,59040,14322664,
      3,6558,14335788,4373,9559378,729<\/strong>,
      1593594,12748510,1062639,13810987,177147,14342320<\/strong>,
      14283325,59031,14302990,19680,14335786,1 }<\/p>\n\n\n\n

      The sixth element can be obtained as 531198==Mod(-27,14342347)^-1 alternatively<\/p>\n\n\n\n


      The twelfth element can be obtained as 14322673==Mod(729,14342347)^-1 alternatively<\/p>\n\n\n\n

      For 6560 from t=8 the 68 elements in ENt17<\/strong> are as shown next.
      { 6560, 43033600, 43044534, 114771574, 729, 4782240,
      124337998, 3188403, 127526239, 531441, 129120454, 128943361,
      177129, 129002392, 59046, 129100798, 1 }<\/p>\n\n\n\n

      Conjecture<\/strong>: The number of elements in the group generated by \u22121+3^t working modulo P where P is 1+(-1+3^t)*(3^(t+1)) must be a multiple of 1+2\u00d7t<\/p>\n\n\n\n

      For 19682 from t=9 the 114 elements in ENt114<\/strong> are not tabulated here
      For 59048 from t=10 the 63 elements in ENt63<\/strong> are not tabulated here
      For 177146 from t=11 the 46 elements in ENt46<\/strong> are not tabulated here
      For 531440 from t=12 the 75 elements in ENt75<\/strong> are not tabulated here
      For 1594322 from t=13 the elements in ENt162<\/strong> are not tabulated here
      For 4782968 from t=14 the elements in ENt29<\/strong> are not tabulated here
      For 14348906 from t=15 the elements in ENt186<\/strong> are not tabulated here
      For 43046720 from t=16 the elements in ENt99<\/strong> are not tabulated here
      For 129140162 from t=17 the elements in ENt70<\/strong> are not tabulated here
      For 387420488 from t=18 the elements in ENt111<\/strong> are not tabulated here
      For 1162261466 from t=19 the elements in ENt234<\/strong> are not tabulated here
      For 3486784400 from t=20 the elements in ENt41<\/strong> are not tabulated here
      <\/p>\n","protected":false},"excerpt":{"rendered":"

      Consider the Proth space We will be using generating element \u22121+3^t The [sub]groups generated by \u22121+3^t have low order [ a lot lower than n-1 ] Tabulating the elements of groups generated by \u22121+3^t next. Using the ENt to avoid clashing with existing letter conventions for groups. Prefer G or S? Replace them in your […]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[5,6],"tags":[7,8],"class_list":["post-141","post","type-post","status-publish","format-standard","hentry","category-mathematics","category-numbertheory","tag-cyclic","tag-group"],"_links":{"self":[{"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/posts\/141","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/comments?post=141"}],"version-history":[{"count":8,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/posts\/141\/revisions"}],"predecessor-version":[{"id":305,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/posts\/141\/revisions\/305"}],"wp:attachment":[{"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/media?parent=141"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/categories?post=141"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/tags?post=141"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}