{"id":160,"date":"2026-08-21T19:07:56","date_gmt":"2026-08-21T19:07:56","guid":{"rendered":"https:\/\/mathrelated.co.uk\/?p=160"},"modified":"2026-09-03T18:39:22","modified_gmt":"2026-09-03T18:39:22","slug":"super-element-3-and-set-1-13t3t1","status":"publish","type":"post","link":"https:\/\/mathrelated.co.uk\/index.php\/2026\/08\/21\/super-element-3-and-set-1-13t3t1\/","title":{"rendered":"Element 3 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 3
<\/p>\n\n\n\n

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

Tabulating the elements of groups generated by 3 next<\/p>\n\n\n\n

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

Conjecture<\/strong>: Order of 3 mod N when N of the form 1+(-1+3^t)*(3^(t+1)) is 6\u00d7(1+2\u00d7t)<\/p>\n\n\n\n

When t=2 we have P=1+8\u00d727 and the set of elements modulo P is a multiplicative group.<\/p>\n\n\n\n


The subgroup generated by 3 has 30 elements<\/p>\n\n\n\n


The elements in ENp30<\/strong> are as shown next
{ 3, 9, 27, 81, 26, 78, 17, 51, 153, 25,
75, 8, 24, 72, -1, 214, 208, 190, 136, 191,
139, 200, 166, 64, 192, 142, 209, 193, 145, 1 }<\/p>\n\n\n\n

If ENp30<\/strong> is really a group then we need inverses so let us document those next.<\/p>\n\n\n\n

    \n
  • Inverse of 3 is 145 mod P<\/li>\n\n\n\n
  • Inverse of 9 is 193 mod P<\/li>\n\n\n\n
  • Inverse of 27 is 209 mod P<\/li>\n\n\n\n
  • Inverse of 81 is 142 mod P<\/li>\n\n\n\n
  • Inverse of 26 is 192 mod P<\/li>\n\n\n\n
  • Inverse of 78 is 64 mod P<\/li>\n\n\n\n
  • Inverse of 17 is 166 mod P<\/li>\n\n\n\n
  • Inverse of 51 is 200 mod P<\/li>\n\n\n\n
  • Inverse of 153 is 139 mod P<\/li>\n\n\n\n
  • Inverse of 25 is 191 mod P<\/li>\n\n\n\n
  • Inverse of 75 is 136 mod P<\/li>\n\n\n\n
  • Inverse of 8 is 190 mod P<\/li>\n\n\n\n
  • Inverse of 24 is 208 mod P<\/li>\n\n\n\n
  • Inverse of 72 is 214 mod P \\\\ 214==Mod((3^14),pr)^-1<\/li>\n<\/ul>\n\n\n\n

    We could alternatively write the group ENp30<\/strong> as follows:
    { 3, 9, 27, 81, (3^5), (3^6), (3^7), (3^8), (3^9), (3^10),
    (3^11), -(3^12), -(3^13), -(3^14), -1,
    -3, -9, -27, -81, -(3^5), -(3^6), -(3^7), -(3^8), -(3^9), -(3^10),
    -(3^11), -(3^12), -(3^13), -(3^14), 1 }<\/p>\n\n\n\n

    From t=3 the 42 elements in ENp42<\/strong> are as shown next
    { 3, 9, 27, 81, 243, 729, 80, 240, 720, 53
    159, 477, 1431, 79, 237, 711, 26, 78, 234, 702,
    -1<\/strong>, 2104, 2098, 2080, 2026, 1864, 1378, 2027, 1867, 1387,
    2054, 1948, 1630, 676, 2028, 1870, 1396, 2081, 2029, 1873,
    1405, 1 }<\/p>\n\n\n\n

    From t=4 the 54 elements in ENp54<\/strong> are as shown next
    { 3, 9, 27, 81, 243, 729, 2187, 6561,
    242, 726, 2178, 6534, 161, 483, 1449, 4347,
    13041, 241, 723, 2169, 6507, 80, 240, 720,
    2160, 6480, -1, 19438, 19432, 19414, 19360, 19198,
    18712, 17254, 12880, 19199, 18715, 17263, 12907, 19280,
    18958, 17992, 15094, 6400, 19200, 18718, 17272, 12934,
    19361, 19201, 18721, 17281, 12961, 1 }<\/p>\n\n\n\n

    From t=5 the 66 elements in ENp66<\/strong> are as shown next.
    { 3, 9, 27, 81, 243, 729, 2187, 6561,
    19683, 59049, 728, 2184, 6552, 19656, 58968, 485,
    1455, 4365, 13095, 39285, 117855, 727, 2181, 6543,
    19629, 58887, 242, 726, 2178, 6534, 19602, 58806,
    -1, 176416, 176410, 176392, 176338, 176176, 175690, 174232,
    169858, 156736, 117370, 175691, 174235, 169867, 156763, 117451,
    175934, 174964, 172054, 163324, 137134, 58564, 175692, 174238,
    169876, 156790, 117532, 176177,
    175693, 174241, 169885, 156817, 117613, 1 }
    <\/p>\n\n\n\n

    From t=6 the 78 elements in ENp78<\/strong> are not tabulated here
    From t=7 the 90 elements in ENp90<\/strong> are not tabulated here
    From t=8 the 102 elements in ENp102<\/strong> are not tabulated here
    From t=9 the 114 elements in ENp114<\/strong> are not tabulated here
    For t=10 the 126 elements in ENp126<\/strong> are not tabulated here
    For t=11 the 138 elements in ENp138<\/strong> are not tabulated here
    For t=12 the 150 elements in ENp150<\/strong> are not tabulated here
    For t=13 the 162 elements in ENp162<\/strong> are not tabulated here
    For t=14 the 174 elements in ENp174<\/strong> are not tabulated here
    For t=15 the 186 elements in ENp186<\/strong> are not tabulated here
    For t=16 the 198 elements in ENp198<\/strong> are not tabulated here
    For t=17 the 210 elements in ENp210<\/strong> are not tabulated here
    For t=18 the 222 elements in ENp222<\/strong> are not tabulated here
    For t=19 the 234 elements in ENp234<\/strong> are not tabulated here
    For t=20 the 246 elements in ENp246<\/strong> are not tabulated here<\/p>\n\n\n\n

    <\/p>\n","protected":false},"excerpt":{"rendered":"

    Consider the Proth space We will be using generating element 3 The [sub]groups generated by 3 have low order [ a lot lower than n-1 ] Tabulating the elements of groups generated by 3 next Using prefix ENp to avoid clashing with existing letter conventions for groups.Prefer G or S? Replace them in your local […]<\/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-160","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\/160","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=160"}],"version-history":[{"count":6,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/posts\/160\/revisions"}],"predecessor-version":[{"id":309,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/posts\/160\/revisions\/309"}],"wp:attachment":[{"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/media?parent=160"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/categories?post=160"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/tags?post=160"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}