{"id":283,"date":"2026-09-03T11:16:05","date_gmt":"2026-09-03T11:16:05","guid":{"rendered":"https:\/\/mathrelated.co.uk\/?p=283"},"modified":"2026-09-03T11:20:26","modified_gmt":"2026-09-03T11:20:26","slug":"element-3-and-set-1-13tx3t","status":"publish","type":"post","link":"https:\/\/mathrelated.co.uk\/index.php\/2026\/09\/03\/element-3-and-set-1-13tx3t\/","title":{"rendered":"Element 3 and set 1+(-1+3^t)\u00d73^t"},"content":{"rendered":"\n

Consider the 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><\/msup>)<\/mo><\/mrow>1+(-1+3^t)*(3^{t})<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

We will be using generating element 3<\/p>\n\n\n\n

The 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 NNe 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) is 6\u00d7t<\/p>\n\n\n\n

When t=2 we have P=1+8\u00d79 and the set of elements modulo P is a multiplicative group.<\/p>\n\n\n\n


The subgroup generated by 3 has 12 elements
<\/p>\n\n\n\n

The elements in NNe12<\/strong> are { 3, 9, 27, 8, 24, -1, 70, 64, 46, 65, 49, 1 }
<\/p>\n\n\n\n

If NNe12<\/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 49 mod P<\/li>\n\n\n\n
  • Inverse of 9 is 65 mod P<\/li>\n\n\n\n
  • Inverse of 27 is 46 mod P<\/li>\n\n\n\n
  • Inverse of 8 is 64 mod P<\/li>\n\n\n\n
  • Inverse of 24 is 70 mod P<\/li>\n<\/ul>\n\n\n\n

    We could alternatively write the group NNe12<\/strong> as follows:
    { 3, 9, 27, (3^4), (3^5), -1,
    -3, -9, -27, -(3^4), -(3^5), 1, }<\/p>\n\n\n\n

    For 703 from t=3 the 18 elements in NNe18<\/strong> are as shown next
    { 3, 9, 27, 81, 243, 26, 78, 234, -1<\/strong>
    700, 694, 676, 622, 460, 677, 625, 469, 1 }<\/p>\n\n\n\n

    For 6481 from t=4 the 24 elements in NNe24<\/strong> are as shown next
    { 3, 9, 27, 81, 243, 729, 2187, 80, 240, 720,
    2160, -1<\/strong>, 6478, 6472, 6454, 6400, 6238, 5752, 4294, 6401,
    6241, 5761, 4321, 1 }<\/p>\n\n\n\n

    For 58807 from t=5 the 30 elements in NNe30<\/strong> are as shown next
    { 3, 9, 27, 81, 243, 729,
    2187, 6561, 19683, 242, 726, 2178,
    6534, 19602, -1<\/strong>, 58804, 58798, 58780,
    58726, 58564, 58078, 56620, 52246, 39124,
    58565, 58081, 56629, 52273, 39205, 1 }<\/p>\n\n\n\n

    For 530713 from t=6 the 36 elements in NNe36<\/strong> are as shown next
    { 3, 9, 27, 81, 243, 729,
    2187, 6561, 19683, 59049, 177147, 728,
    2184, 6552, 19656, 58968, 176904, -1<\/strong>,
    530710, 530704, 530686, 530632, 530470, 529984,
    528526, 524152, 511030, 471664, 353566, 529985,
    528529, 524161, 511057, 471745, 353809, 1 }<\/p>\n\n\n\n

    For 4780783 from t=7 the 42 elements in NNe42<\/strong> are as shown next
    { 3, 9, 27, 81, 243, 729,
    2187, 6561, 19683, 59049, 177147, 531441,
    1594323, 2186, 6558, 19674, 59022, 177066,
    531198, 1593594, -1<\/strong>, 4780780, 4780774, 4780756,
    4780702, 4780540, 4780054, 4778596, 4774222, 4761100,
    4721734, 4603636, 4249342, 3186460, 4778597, 4774225,
    4761109, 4721761, 4603717, 4249585, 3187189, 1 }<\/p>\n\n\n\n

    For t=8 the 48 elements in NNe48<\/strong> are not tabulated here
    For t=9 the 54 elements in NNe54<\/strong> are not tabulated here
    For t=10 the 60 elements in NNe60<\/strong> are not tabulated here
    For t=11 the 66 elements in NNe66<\/strong> are not tabulated here
    For t=12 the 72 elements in NNe72<\/strong> are not tabulated here
    For t=13 the 78 elements in NNe78<\/strong> are not tabulated here
    For t=14 the 84 elements in NNe84<\/strong> are not tabulated here
    For t=15 the 90 elements in NNe90<\/strong> are not tabulated here
    For t=16 the 96 elements in NNe96<\/strong> are not tabulated here
    For t=17 the 102 elements in NNe102<\/strong> are not tabulated here
    For t=18 the 108 elements in NNe108<\/strong> are not tabulated here
    For t=19 the 114 elements in NNe114<\/strong> are not tabulated here
    For t=20 the 120 elements in NNe120<\/strong> are not tabulated here
    <\/p>\n\n\n\n

    Using sturdy element notation we might say the results tabulated in this post are<\/p>\n\n\n\n

    \"Sturdy<\/figure>\n\n\n\n

    <\/p>\n\n\n\n

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

    Consider the space We will be using generating element 3 The groups generated by 3 have low order [ a lot lower than n-1 ] Tabulating the elements of groups generated by 3 next Using prefix NNe to avoid clashing with existing letter conventions for groups.Prefer G or S? Replace them in your local copy. […]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[12,5],"tags":[7,8],"class_list":["post-283","post","type-post","status-publish","format-standard","hentry","category-abstractalgebra","category-mathematics","tag-cyclic","tag-group"],"_links":{"self":[{"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/posts\/283","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=283"}],"version-history":[{"count":2,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/posts\/283\/revisions"}],"predecessor-version":[{"id":288,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/posts\/283\/revisions\/288"}],"wp:attachment":[{"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/media?parent=283"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/categories?post=283"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/tags?post=283"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}