{"id":148,"date":"2026-08-20T20:01:40","date_gmt":"2026-08-20T20:01:40","guid":{"rendered":"https:\/\/mathrelated.co.uk\/?p=148"},"modified":"2026-09-03T18:38:43","modified_gmt":"2026-09-03T18:38:43","slug":"super-element-13t-and-set-113t3t1","status":"publish","type":"post","link":"https:\/\/mathrelated.co.uk\/index.php\/2026\/08\/20\/super-element-13t-and-set-113t3t1\/","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>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 1+3^t<\/p>\n\n\n\n

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

Tabulating the elements of groups generated by 1+3^t next<\/p>\n\n\n\n

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

When t=2 we have P=1+10\u00d727 and the set of elements modulo P is a multiplicative group.<\/p>\n\n\n\n

Our generator 1+3^t is 10 and the subset it generates has 5 elements.<\/p>\n\n\n\n

The elements in EPu5<\/strong> are { 10, 100, 187, 244, 1 }<\/p>\n\n\n\n

If EPu5<\/strong> is really a group then we need inverses so let us document those next.<\/p>\n\n\n\n

    \n
  • Inverse of 10 is 244 mod P<\/li>\n\n\n\n
  • Inverse of 100 is 187 mod P<\/li>\n<\/ul>\n\n\n\n

    For 28 from t= 3 the elements in EPu42<\/strong> are as shown next.
    { 28, 784, 1531, 2026, 3, 84<\/strong>, 83, 55, 1540, 9,
    252, 249, 165, 82, 27, 756, 747, 495<\/strong>, 246, 81,
    -1, 2241, 1485, 738, 243, 2266, 2185, 2186, 2214, 729<\/strong>,
    2260, 2017, 2020, 2104, 2187, 2242<\/strong>, 1513, 1522, 1774, 2023,
    2188, 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

    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 82 from t=4 the 27 elements in EPu27<\/strong> are shown next.
      { 82, 6274, 13339<\/strong>, 17740, 9, 738, 735, 489, 244,
      81, 6642, 6615, 4401, 2196, 729<\/strong>, 19924, 19681, 19682,
      19764, 6561, 19900<\/strong>, 17713, 17722, 18460, 19195, 19684, 1 }<\/p>\n\n\n\n

      Conjecture<\/strong>: The number of elements in the group generated by 1+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 244 from t=5 the 22 elements in EPu22<\/strong> are not tabulated here
      For 730 from t=6 the 39 elements in EPu39<\/strong> are not tabulated here
      For 2188 from t=7 the 90 elements in EPu90<\/strong> are not tabulated here
      For 6562 from t=8 the 17 elements in EPu17<\/strong> are not tabulated here
      For 19682 from t=9 the 114 elements in EPu114<\/strong> are not tabulated here
      For 59048 from t=10 the 63 elements in EPu63<\/strong> are not tabulated here
      For 177146 from t=11 the 46 elements in EPu46<\/strong> are not tabulated here
      For 531440 from t=12 the 75 elements in EPu75<\/strong> are not tabulated here
      For 1594322 from t=13 the elements in EPu162<\/strong> are not tabulated here
      For 4782968 from t=14 the elements in EPu29<\/strong> are not tabulated here
      For 14348906 from t=15 the elements in EPu186<\/strong> are not tabulated here
      For 43046720 from t=16 the elements in EPu99<\/strong> are not tabulated here
      For 129140162 from t=17 the elements in EPu70<\/strong> are not tabulated here
      For 387420488 from t=18 the elements in EPu111<\/strong> are not tabulated here
      For 1162261466 from t=19 the elements in EPu234<\/strong> are not tabulated here
      For 3486784400 from t=20 the elements in EPu41<\/strong> are not tabulated here<\/p>\n","protected":false},"excerpt":{"rendered":"

      Consider the Proth space We will be using generating element 1+3^t The [sub]groups generated by 1+3^t have low order [ a lot lower than n-1 ] Tabulating the elements of groups generated by 1+3^t next Using the prefix EPu 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-148","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\/148","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=148"}],"version-history":[{"count":6,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/posts\/148\/revisions"}],"predecessor-version":[{"id":306,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/posts\/148\/revisions\/306"}],"wp:attachment":[{"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/media?parent=148"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/categories?post=148"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/tags?post=148"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}