{"id":164,"date":"2026-08-21T19:36:06","date_gmt":"2026-08-21T19:36:06","guid":{"rendered":"https:\/\/mathrelated.co.uk\/?p=164"},"modified":"2026-09-03T18:39:37","modified_gmt":"2026-09-03T18:39:37","slug":"super-element-3-and-set-113t3t1","status":"publish","type":"post","link":"https:\/\/mathrelated.co.uk\/index.php\/2026\/08\/21\/super-element-3-and-set-113t3t1\/","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>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 the prefix EPr 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+10\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 EPr30<\/strong> are as shown next
{ 3, 9, 27, 81, 243, 187, 19, 57, 171, 242
184, 10, 30, 90, -1<\/strong>, 268, 262, 244, 190, 28,
84, 252, 214, 100, 29, 87, 261, 241, 181, 1 }
<\/p>\n\n\n\n

If EPr30<\/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 181 mod P<\/li>\n\n\n\n
  • Inverse of 9 is 241 mod P<\/li>\n\n\n\n
  • Inverse of 27 is 261 mod P<\/li>\n\n\n\n
  • Inverse of 81 is 87 mod P<\/li>\n\n\n\n
  • Inverse of 243 is 29 mod P<\/li>\n\n\n\n
  • Inverse of 187 Is 100 mod P<\/li>\n\n\n\n
  • Inverse of 19 is 214 mod P<\/li>\n\n\n\n
  • Inverse of 57 is 252 mod P<\/li>\n\n\n\n
  • Inverse of 171 is 84 mod P<\/li>\n\n\n\n
  • Inverse of 242 is 28 mod P<\/li>\n\n\n\n
  • Inverse of 184 is 190 mod P<\/li>\n\n\n\n
  • Inverse of 10 is 244 mod P<\/li>\n\n\n\n
  • Inverse of 30 is 262 mod P<\/li>\n\n\n\n
  • Inverse of 90 is 268 mod P<\/li>\n<\/ul>\n\n\n\n

    We could alternatively write the group EPr30<\/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 EPr42<\/strong> are as shown next
    { 3, 9, 27, 81, 243, 729, 2187, 2023, 1531, 55,
    165, 495, 1485, 2186, 2020, 1522, 28, 84, 252, 756,
    -1<\/strong>, 2266, 2260, 2242, 2188, 2026, 1540, 82, 246, 738,
    2214, 2104, 1774, 784, 83, 249, 747, 2241, 2185, 2017,
    1513, 1 }<\/p>\n\n\n\n


    From t=4 the 54 elements in EPr54<\/strong> are as shown next
    { 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 19195,
    17731, 13339, 163, 489, 1467, 4401, 13203, 19682, 19192, 17722,
    13312, 82, 246, 738, 2214, 6642, -1<\/strong>, 19924, 19918, 19900,
    19846, 19684, 19198, 17740, 13366, 244, 732, 2196, 6588, 19764,
    19438, 18460, 15526, 6724, 245, 735, 2205, 6615, 19845, 19681,
    19189, 17713, 13285, 1 }<\/p>\n\n\n\n


    From t=5 the 66 elements in EPr66<\/strong> are as shown next.
    { 3, 9, 27, 81, 243, 729,
    2187, 6561, 19683, 59049, 177147, 175687,
    171307, 158167, 118747, 487, 1461, 4383,
    13149, 39447, 118341, 177146, 175684, 171298,
    158140, 118666, 244, 732, 2196, 6588,
    19764, 59292, -1<\/strong>, 177874, 177868, 177850,
    177796, 177634, 177148, 175690, 171316, 158194,
    118828, 730, 2190, 6570, 19710, 59130,
    177390, 176416, 173494, 164728, 138430, 59536,
    721, 2193, 6579, 19737, 59211, 177633,
    177145, 175681, 171289, 158113, 118585, 1 }<\/p>\n\n\n\n


    From t=6 the 78 elements in EPr78<\/strong> are not tabulated here
    From t=7 the 90 elements in EPr90<\/strong> are not tabulated here
    From t=8 the 102 elements in EPr102<\/strong> are not tabulated here
    From t=9 the 114 elements in EPr114<\/strong> are not tabulated here
    For t=10 the 126 elements in EPr126<\/strong> are not tabulated here
    For t=11 the 138 elements in EPr138<\/strong> are not tabulated here
    For t=12 the 150 elements in EPr150<\/strong> are not tabulated here
    For t=13 the 162 elements in EPr162<\/strong> are not tabulated here
    For t=14 the 174 elements in EPr174<\/strong> are not tabulated here
    For t=15 the 186 elements in EPr186<\/strong> are not tabulated here
    For t=16 the 198 elements in EPr198<\/strong> are not tabulated here
    For t=17 the 210 elements in EPr210<\/strong> are not tabulated here
    For t=18 the 222 elements in EPr222<\/strong> are not tabulated here
    For t=19 the 234 elements in EPr234<\/strong> are not tabulated here
    For t=20 the 246 elements in EPr246<\/strong> are not tabulated here<\/p>\n\n\n\n

    Using sturdy element notation we might say that the groups tabulated here are<\/p>\n\n\n\n

    \"Super<\/figure>\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 the prefix EPr 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-164","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\/164","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=164"}],"version-history":[{"count":5,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/posts\/164\/revisions"}],"predecessor-version":[{"id":310,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/posts\/164\/revisions\/310"}],"wp:attachment":[{"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/media?parent=164"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/categories?post=164"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/tags?post=164"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}